Sounds like a forum would suit your needs--IP.B is the best social forum system, with status updates and other social features. xenForo is a good alternative to IP.B. If you're looking for free software I suggest MyBB as your forum, as it is, in my opinion, the best open-source forum system.
For private, group based discussions, you could have group-specific forums, which are only visible to group members, as well as a open forum so members of all groups can talk together. Any of the forum softwares mentioned above could do this.