BuddyPress.org

You could create one group that all people join when they register.

Use DJPaul’s Welcome Pack plugin for that:

https://wordpress.org/extend/plugins/welcome-pack/

Otherwise go the external bbPress route, and yes, you’ll need to theme it.

[EDIT]

Forgot that Paul changed his plugin so it sends out group invites, instead of auto joining groups.

If you want to make Paul’s plugin automatically join groups.

Change the following lines in the Welcome Pack plugin from:

groups_invite_user( array( 'user_id' => $new_user_id, 'group_id' => $group_id, 'inviter_id' => $group->creator_id ) );

groups_send_invites( $group->creator_id, $group_id );

to:

groups_join_group( $group_id, $new_user_id );

I should say I haven’t tried this, but it should work.