Not sure how to do that with Moodle chat, but....
BIG DISCLAIMER: I HAVEN'T TRIED THIS IN A REAL CLASS. USE AT YOUR OWN RISK.
You may get something like what I'm testing at the link below to work.
http://www.extremeclassroom.com/lms18/
You would need to edit your theme and widen the left or right block a bit...about 350px wide should do the trick...I haven't done that. From my limited experimenting it looks like no accounts are needed. I'm not sure if you can make them completely private, but you can set them so they aren't listed and where the messages are not saved.
Create them here: http://xat.com/
Steve