The 'no chatting during the pomodoro rule' shouldn't be too strict in my opinion. I think that most people are alright if there is some minimum talking during pomodoro times especially for relevant things like this or greeting people, am I wrong? In addition, immediate feedback has been shown to facilitate learning thus scolding the person immediately as you see them slacking off should help to change their behavior faster.

There is no need of any notifications of the chat unless you want them as well. Private messaging is also a desired feature but in my opinion it is better (but not neccessery) if everything related to the task at hand is out in the open in order to create a 'community' atmosphere.