Bridge private rooms to Matrix

For reference, DM rooms are already supported which are private rooms between two people, #2671 (closed)

Considerations

How do you invite a Matrix user to a private Gitter room?
A Matrix user won't appear in the Gitter people roster so you might not know someone is lurking who shouldn't be
- #2688 tracks adding Matrix users to the people roster but we were thinking of just doing the simple approach and only adding people who recently chatted in the visible chunk of messages. This doesn't help the situation of not knowing about lurkers.
After Gitter is sunset, how does the same person on Matrix re-gain access to the same rooms they could with their Gitter account?