Why not just invite each other to a table, or make a private analysis board?
Private Chess Rooms

if 2 people wanted to play only each other
You mean, like some way of finding, messaging, and challenging another player, and playing a game with them where you have the ability to chat, etc.? Oh, wait.
Private rooms are for gentlemen's clubs and dubious internet social sites ;).

Private rooms is Yahoo.com. This is Chess.com.
Yahoo is a disaster for playing chess.
So no to the private rooms.

Suppose I have a friend in Chicago and I live in NY. We know that we both will be available on the internet at 8:00 PM. Would it be possible for us to contact each other and play chess together without putting out a general seek and having strangers click on it?
I was thinking that if 2 people wanted to play only each other it would be easy to do if a chess website such as chess.com would make available private rooms. This would be a feature that no other website has. Would it be possible to do this?