Two things:
1) You are able to have private chats with people within chat. You just click on their name on the right side in chat and select private chat (or something like that) and then it pops up a private window that you can talk to another person without anyone else reading it.
2) The chat bar at the bottom of the screen had a lot of bugs. If you wrote in it and then went to another page it seemed to cause problems and you wouldn't be able to see the other person's reply. Even if you opened another window it still had the same problems. It was annoying to use because of this.