Chess is a game of complete information. You can look at the board and know what's going on. Sometimes you might need to know if one side has moved their King or not, to know if it is legal to castle. But in that case, you could just look at the score of the game. Bughouse is almost like that. A strong player can watch their partner's board and see exactly which pieces will help, and hurt. But bughouse is not quite a game of complete information. Take the following example. I've made it up, but this sort of thing happens a lot.
Here White can mate by dropping a heavy on h8. White can also take the Rook on a8. Now look at White's partner's board.
Black can play Nc2 and force get a Rook for their partner, which wins. Or Black can wait for White to sacrifice Queen for Rook, which Black can drop at h1. If you're reasonably perceptive, you can see why the example is the way it is. There is a lot of potential for the team to mess up, and mate themselves. If both players feed one another, White could be mated by N@e2. Black could be mated by the Queen on g7. If neither player feeds, or if they hesitate for too long, both players will be down on the clock with positions which are difficult, if not impossible, to defend.
Even supposing there were strong bughouse engines, they wouldn't be able to tell you which player should sacrifice for the other, and which should sit, because bughouse is not quite a game of complete information. Hopefully this illustrates the need to coordinate. Many reasonably strong players play bughouse as it is a two-against-two competition of crazyhouse--which is a bad mistake. The whole point of bughouse is the relationship between the boards.
Let's make a mental note of this.
#1. Coordination is strictly necessary
I tried to construct the above example so that White and Black were in essentially the same situation. Sometimes this happens. But sometimes, you get scenarios like the following.
White is busted. Apart from a few checks, White is completely out of resources either for attack or defense. Although White happens to have mate with a Rook. What is White's partner doing?
Black's board is as congested as White's board is sparse. Black should be urgently trying to feed White some reinforcements. It would not be very hard for Black to force a Rook so that White can place it on b4, mating instantly. It happens frequently in games like this that both White and Black are urgently requesting trades. White wants the Rook, which is the simplest way to win. Black wants a Knight to put on f3/g2. Who is right? Obviously, White is more in need of trades. Black needs a Knight to mate, but Black isn't in trouble if it isn't coming. On the other hand, White is in trouble without the Rook. While White can take the Knight, this allows White's opponent to promote pawns to Queens, with check. So it should be obvious who should get the trades. Black should play for White here. I think this is worth enshrining in bold.
#2. Prioritize the board which is more empty
I've often talked about sac sitting, which for anyone who has played bughouse for less than 8 seconds. looks something like this.
As an occasional weapon, this can work. Every game, it is downright bad. And it isn't a question of style. Style is a choice between equally good options. Neglecting your development and demanding high flow not only makes your own position worse, but it constrains your partner. Now they can't focus on outplaying their opponent directly, but they are forced to play for their partner right out of the opening--all the while down material. Volumes could be written on the wrongness of this approach. In fact I am working with Everyman Chess to produce a 9 volume series on the wrongness of early sac-sitting in bughouse. The first installment will come out early next year.
But how about defense? Curiously, if you copy the style of a sac sitter, your prospects get worse, because your team is doubly down material. But if you copy the style of the player who invites the sacrifice, your prospects improve. Instead of being doubly down material, you will be doubly up material. However, even with this being the case, both players will be a little exposed. Consider the following scenario. I will just describe it, rather than setting up any position.
Both you and your teammate are up a lot of material, and you are both up on the clock. This means that you can always wait. If either of your opponents pauses their attack to trade for the other board, and stops giving check, then that player can counterattack. So if your opponent stops checking you to feed their partner, you can start to use your extra material. Sometimes, what's required is for the a team to sit and defend in alternation. If either play stops checking to take material for the other board, you counterattack, and maybe even have your partner sit for the rest of the game while you mate.
I have seen so many teams fail to do this, which is the most simple and straightforward way to punish a team of sacsitters, that I think it is worth enshrining in bold.
#3. Take turns sitting and defending.
It's oddly specific, maybe, but true nonetheless. Of course, it's easier to communicate quickly with the necessary detail and nuance if you are using voice. But if you realize that taking turns sitting and defending is a real and important technique, you will be ready for it when you encounter the need for it in your own games. In a few games I have observed, I have seen players despairing when they are actually winning. "Why aren't you safe?" or, "Why do you trade so much?" Locally, these complaints seem warranted. One board considered in isolation, one player may be unsafe, and another player may have traded too much. But actually "one board considered in isolation" makes no sense, because bughouse is played on two boards. Global assessments of the game are always truer.
The next point is about trades. I think that every player has heard the phrase "to take is a mistake".
This is bughouse! You're supposed to trade, no? In fact, White helps Black by taking on d6, just as Black helps White by taking on g3. If White takes d6, Black retakes with the pawn, controlling e5. If Black takes g3, White takes with the h pawn, developing the Rook, and cementing control of e5, since the only pieces controlling that square are now White's. Of course, if White is a team player, and White's teammate asks for trades, White should set aside their own positional ambitions, perhaps. But it is worth noting that even materially equivalent trades can be positionally costly. Sometimes trading the house can cost you your position completely. Take, for instance...
You've done well for your partner, you are up the exchange, and you traded everything for your partner, who has a very promising attack. You are optimistic until your opponent drops a pawn at e2. You move the rook to b1, and your opponent drops another pawn at d2. Now your King is on the run. It isn't easy to grab trades and be immune from this sort of thing. Asking that players trade high, quickly, and safely, is not always reasonable or fair.
#4. High trades can be positionally costly
So where does this leave the aspiring bughouse wizard, who likes to win by crashing through with sacrificial attacks? Surely you can't be expected to play chess, especially when there are titled players roaming around the random pool. If you don't have high flow, you are surely going to be doomed as they grind you down in a rook and pawn endgame. Actually, not so. You can be strong at bughouse without being strong at chess. Many players are. And they manage decently with low flow. The difference between chess and bughouse is one of degree. As a thought experiment, imagine that you are playing against a titled player, maybe an FM or an NM. You can drop pieces, and they cannot. How much do you need to win? If you are strong, you could probably get by with just a pawn, if you can drop it at any time. Any endgame would be in your favor. If you are like me, you might need a piece or more. And yet strong bughouse players can outplay titled chess players in bughouse--they do it routinely--with low flow. I think that this is key.
#5. You can play without trades--without being a great chess player
Perhaps this sounds unbelievable. But in bughouse you are dealing not just with the position, but with future drops. And a strong bughouse player can play for those future drops without being Carlsen-level strength in chess. And this is how you get out of the false dichotomy which says that the game is either high trades all game or it's just standard chess. Strong bughouse players can do a lot with just pawns and minor pieces.
If I'd have designed the bughouse partner tells, I might have had at least two tells for trades. The first would be "trades are nice, but not necessary". The second would be "trades are essential to keep us in the game". And conversely, I might have a few tells for low trades. Firstly, "any piece mates me", secondly, "hold a blocker, except for pawns", for situations where there is a chessmate on the back rank, thirdly "hold all blockers", and fourthly, "low flow is nice".
A short addendum. To summarize, if your partner is being sacked on, it pays to also get sacked on. But if your partner is sacking, you have to (somehow) play to feed your partner. This tells you what to do based on what your partner is doing, but there are also a couple of useful rules of thumb based on what your opponents are doing.
If you are stronger than your opponent, you want a smooth, reliable path to victory. You want to minimize risks, and you want to win all or most of the games you play. For that, less volatile openings make more sense. 1.d4 will allow you to stay in control as White while continuously pressuring your opponent without much risk to yourself or to your partner. (For this, you can see my post on the bughouse London System.) As Black, perhaps a French Defense.
If you are playing against someone who is stronger, then you want to disrupt the status quo. It's in your interests to rock the boat. A nice opening for doing this with Black, which I don't specialize in but have used to beat several players who are much stronger than me, is the following.
It's a messy variation. But when playing against 2700-2800s, chaos is your friend. You will need the flow to be in your favor, but if you are playing against a 2700 rated player, in the random pool, it might be reasonable to expect the flow to be in your favor.
However, it would be a strategic mistake to employ this sort of thing routinely against weaker players. Why would you go in for this if you can outplay someone consistently in more usual lines?
It might be considered odd, or at least impure, to strategize against the player. The Grandmaster Gligoric wrote a book called I Play Against Pieces, which seems to imply that he did not play differently according to his opponent. The idea of looking for the objectively best moves seems incoherent, for the following reason. If we knew what the objectively best moves were, we would need a deeper game. Objectively, there are only moves which lead to a win, draw, or loss. "Strong" and "weak" are evaluations you make when the truth is hidden. When we try for "tricks", we are always trying to exploit the limited vision of our opponents, whether they are humans or machines. Some threats are more well concealed than others, but only for players who have limited vision. The idea of making a move which contains a subtle threat, rather than a superficial one, is only possible when the game has secrets. Tic-tac-toe has no secrets from anyone who has played more than five games of it. All threats are equally superficial in an objective sense, and in a subjective sense. But in chess (or in any game of complete information), all threats are superficial in an objective sense--there is no strong and weak to any player capable of seeing everything. But it is the subjective sense that makes the game interesting--and chess is sufficiently deep for human beings--for now.
Hopefully this will disabuse you of the idea of searching for perfect objectivity in your games. The job is to make things as difficult for your opponent as you can, and this requires having some sense of what your opponent can see. And so you should always be ready to muddy the waters when faced with opponents who know lots of theory, and who thrive in the more usual structures.
A second addendum, this one much shorter: when coordinating with your partner, you should first pay attention to the colors. It is a little harder to defend as Black. It is easier to deal with a wayward Queen as White, then as Black. Black frequently will give up a Queen for two minor pieces, and Black's partner has the White pieces, and usually manages with it. If your partner is playing Black, you should take more care about which pieces you give, and when.
Secondly, who has the stronger opponent? If you are on the lower board, and your partner is playing against a 2600 rated player, you might reasonably be asked to indulge your partner a few lost Queens, and a few long sits to find key moves. You should probably not look for forcing lines, but for ways to persistently pressure your opponent, giving them many options and many chances to mess up. And you should not make high demands on a partner who plays someone much stronger than themselves.
Hopefully this helps.