S-3 NFOs that flew in the right seat usually got to drive when things were slow or on FACIT (Fool Around and Call It Tactics) missions. When you were actually working on a mission you had other things to do in the right seat. Besides, the flying being done at those times was better left to the guy with the single anchor wings. That is his job.
All that said, it doesn't make any difference as the War Hoover is just about gone. You still might want to take this on board. No self respecting NFO "asks" for the stick. You have your job, the pilot has his. I was an experienced civ pilot as a S-3 NFO and an airline pilot by the time I flew in the Reserves. I never once asked for the stick. Got to drive plenty, but only when offered by the pilot in command.