There's a good existing answer on a basic schema in SQL.
I can understand it, it's very straightforward. We have a user table, a chat table, and a chat_line table (which in a sane world would be called messages).
I'm rather new to NoSQL, and my mind is still used to "normal" SQL schemes, and I'm trying to understand what would be the correct schema for a chat app in NoSQL (like mongo, or whathaveyou).
I'm talking the simplest form, between one user to another, nothing special - no file messages, no pictures, no group chats. Just text.