Sorry, maybe its mentioned elsewhere, but do you gain any faction in private matches? I'd imagine not, but would like to know for sure!
30rep, more if you have on a booster.
I don't think crucible boosters will work in private matches, unfortunately.
- All participating players will receive 30 Crucible Reputation upon completing each Private Match. This can be increased with the Crucible Champion perk found of some Class Items, however Crucible Reputation Boosters will have no effect.
https://www.bungie.net/7_Destiny-Rise-of-Iron-Private-Matches/en/Help/Troubleshoot?oid=45079