People can post their MSN or Yahoo account name, and they can make all sorts of weird comments about the teacher behind his/her back.
I don't quite see the point in preserving a false image that students won't be able to communicate when they can.
I can understand and appreciate personal info like email addresses or phone numbers being hidden, but if I am teaching English (or any other subject using English) and 2 people come in, one might not understand me because their English level isn't good enough to enjoy the lesson.
If a friend came with them and acted as a translator, then they could talk privately without it interrupting the main chat area (text will be flying and hard for the teacher to keep up with if all comments can only be put in one window). Having to sign on to MSN or Yahoo to achieve this is too cumbersome. Also, it uses up computer resources.
To recap:
1. People can already message each other by using another messaging program like MSN or Yahoo.
2. Students can help each other out while the teacher continues their lesson without interruptions.
3. There is a lot of text in chat, but if they could private chat then there wouldn't be as much. Only important comments could be made and this would be easier for the teacher to monitor.
Another idea is that when I teach, sometimes I like to make games using vocabulary or something else. Students can team up and private message without another team seeing their answers.
And, my mind is still coming up with reasons, so I'll mention yet another one. If the teacher mutes text then no one can communicate. If sound stops working or some other technical difficulty occurs, then the students have no way to communicate this.
This can also be used with screening interview questions. People could PM a student who screens questions. Those accepted would be posted in chat. Since we are worried about inappropriate content, then wouldn't this help in filtering it out?