Private exchanges work pretty much the same as other Throne exchanges, only with private exchanges you can keep it to your community with a custom URL for the exchange!
You can create a private exchange here.
There is also rematching for private exchanges!
It works very similarly to public exchanges.
Private exchange participants can register as a re-matcher and then if someone does not choose a gift, for whatever reason, they will be given to a re-matcher who will then choose a gift!
If the pool of registered re-matchers is all used then all remaining users who have still not received a gift will be re-matched with the creator of the exchange!