No priority. Sometimes your companion will be given a break by the gate agent, sometimes not. We once boarded after about 10 standbys, but we made the flight.
Then what happens if the A-lister (Companion Pass holder in this case) clears and the companion doesn't because there are not enough seats available on the flight? Does the companion have to pay full fare to get to their destination because they aren't on the flight with the Companion Pass holder? Or do they not clear either of them unless there are enough seats for all standbys?