BuddyPress.org

Its been so long to your post , but I wonder where have you reached with your idea.
I am looking for something similar, but not just replacing the name from group to project. Can there be any drawing tool, so that we can just draw and share.?

anyways if somebody come across again,this plugin can be helpful :

https://wordpress.org/extend/plugins/buddypress-group-documents/

for sharing documents inside a group.