I would say not a whole lot short of master level.
It's more important to understand the ideas behind the openings rather than memorizing lines.
Your opponent will inevitably deviate from the memorized line 5-10 moves in and if you understand why those moves are played then you will understand why their deviation is weak and how to take advantage of it.
If you just have memorized lines then you will just be confused.
Having said that, some memorization skills will help greatly. They're just not required.
So, I'm a beginner (less than one year of exp.) and have a rating of 900-something online here at Chess.com (I don't know exact rating, b/c haven't checked and lazy right now)....please don't laugh! I'm still learning.
My question, however, is how much memory is needed to improve in chess?
I'm a beginner now and have literally memorized close to nothing in chess aside from two openings for WHITE and one opening each as a response to e4 and d4 white openings for BLACK. And when I say I've "memorized" these, I mean the first 7 or 8 moves of those openings and nothing that enters the middle game.
I figured that would be the absolute bare minimum needed. It's worked OK for me so far as I'm at least able to get into a middle-game without being mated lol (as I was in the past).
But, I know there are many more openings for both W and B and so many variations of them ....
i.) At what point are memory skills required and kick in in order to succeed?
ii.) Is memory mostly only for openings or does it apply to the middle game too (in which case it would seem endless, given how many possibilities there exist there)?
iii.) What type of memory skills are we talking about? ...Weird question...I know. But it's interesting to me, b/c chess is a visual game and I wonder if anyone literally just memorizes lines in algebraic notation only (and then plugs them into the game visually later) or if they memorize these openings visually in their mind?

Feel free to add more comments on other tangential topics as well. Your feedback is appreciated. Discussion would be interesting too (feel free to share own exp. too).
TVM!!!