If I understand correctly, you’re talking about the list of categories shown in the user’s sidebar?
There is a site setting for default sidebar categories:
If you add the private category there, it should show up in people’s sidebars once they join the group and disappear when they leave the group. There may be some edge cases where it needs to get added manually if they have edited their own sidebar previously.
As users are able to add/remove things themselves, it may also be worth having a topic on your site to refer to in case people need help doing that.