It would take a bit of a change to the layout of ChattyWays as it currently is. Well, one thought would be to start a new chat with the users that you select. That sound a bit like I am restating the obvious, so let me cover in more detail. For a private chat, you can probably select the users that are currently involved in the chat and then you could just create a new form that allows you to send a message to just that user. In the main chat, messages are retrieved based on the current chat and last message id. For a private chat, you could add additional filtering based on the users involved in the private chat and even based the private chat form on the main chat form. I am really not familiar with Yahoo's Private chat functionality so I am going based on your description.
Why don't you let me know if you have made any progress with it and we can take it step by step. This definitely seems doable and not really that difficult. I will try to setup Chattyways on Community Credit so that we might be able to do a bit of experimentation on new features.