Can I create a new chat room?

No.
If you want to privately say something to someone, leave them a comment.
Go to their profile page and there’s a textbox for you to enter your message.

There are other chat rooms that you can have someone meet you in just pick an empty one, but it’s not completely private.

Thanks!

You can also, if you are close efriends, simply swap emails.

It is possible to create chat rooms in general, but to create one for fluther users would be a challenge.