How can you get to talk with another person you're having a problem with? This is to solve the problem of course and also to make sure there isn't any unnecessary down voting on random questions and answers.
Search for the user in the chat user page, and then go to their chat profile. Once you're there click on the "start a new room with this user" button. That will create a chat room and should send the user an invitation to join the room.
To add on to what @wipqozn said, only moderators have the ability to create truly invisible chat rooms that only people invited (as well as any other moderators in the Stack Exchange network) can read and write in it. Note that a normal user like yourself can create gallery chat rooms, and while others may not be able to write in it, they can certainly read it. Beware of what you say in there.
Also, emphasising what @GraceNote said below, this is for