[plug in: chat room] Looking for a chat room plug-in with log-in option

We are seeking advice.
We are a college senate running a web page on the college wordpress site.
We are looking for a wordpress ‘chat room’ (not a live chat) plug-in with a login so web page visitors don’t see the conversation(s) going on between senate members. Thus, conversations are private seen only by senate members with login passwords.
We are looking for recommendations. Is there a free plug-in that can do this?
Thanks.