I believe that you should have a setting that, when turned on, will send you a notification that one of your chat friends has joined the chat. And, if possible, it would be great is you could set it so that you only get notified when specific friends join. After writing this I finally became aware that the chat will display a desktop notification for this, but I was suggesting an in chat notification so that people on mobile devices can also get it.

Edit: I'm not talking about a mobile push notification, I'm just talking about having the chat whisper to you or something that one of your friends has joined the chat.