Carlo Gambino was on the Hit squad of Albert Anastasia, not setting it up as most captains do but he was on the actual hit squad, there with a .38 on the hit squad. Unless he was designated the back up shooter like in some hits then im certain he pulled the trigger as being on the hit squad and not pulling the trigger gets you labeled many times as a Rat. Thats why you always see in the movies the second guy pumping the "good measure" shots into the dead body laying at his feet. Same philosiphy as "making bones" and undercover cop or informant cannot kill.

As for Gotti, no one REALLY knows if he has ever pulled the trigger, it is also known that Gotti never made his bones. He moved up very rapidly when Carmine Fatico got pinched, at 33 Gotti was named acting captain only like 6 months after he was made.