That's already the case if you're viewing their profile or another users profile. If you mean on the Recent Activity page then no it will always display any activity they've access to. There is no private groups except those marked as Invite type and it should not be displaying Invite group activity unless they are a member of that group or are a CB Moderator.

Is there any way to only display group activity for groups that are members has "joined" even if it's open group?

No.

Is there a way to filter like that?

The only way to filter activity down is using the Asset, which there isn't one specifically for filtering down to joined. It's possible to filter it down to only GJ activity, but it will show all GJ activity the viewing user has access to.

Once filters are actually implemented, which will just let users select a filter from a dropdown and it will filter the asset of the stream, this maybe possible. There's also plans for an exclude asset parameter for streams as well.

Approval groups content is not hidden. The only difference between Open and Approval is one requires approval to join. This may change in a future release when more group types are added.