I've set up freichat here. It's J2.5.7 with CB 1.8.1.
It shows the correct avatar in the list of users, but if I start a chat with another user, it doesn't show my name at the top of the chat, and when I say something it shows "Me:" at both ends, and not the name in the other end.

.If you have phpmyadmin , open you database , and open the table frei_chat .
Now , check the fields from_name and to_name in the inserted records in that table , Are they same or different ?

from_name and to_name are both empty. The from and to contains the correct id's for the users.

Could it be that you use the CB field firstname? That's empty here, I only have fullname for my users.

In that case change the setting in the backend to show username of the user instead of his full name .

Ok, so fullname isn't really full name or 'name', it's firstname? Changing it to username made it show the correct name in the list.
Not all now each others usernames, so I would like for it to show the name instead, can that be changed in a future version?

Not sure what I did wrong then, I see:
1. username
2. nickname
3. fullname
I'm pretty sure that I selected Joomla with CB as the installation type.

That is fine , choose nickname .

That is fine , choose nickname .

Ahh, yes, that made the name appear, excellent, thankyou!