Regardless whether your ball is on the green, the player further from the hole has the honor (i.e. you have the honor even though you're on the green). You'll see this a bunch on the PGA telecasts.
Most of the time, though, we'll usually let the guy on the fringe play first. That way you don't have to take the pin out for the guy on the green and then potentially replace it for the guy on the fringe.
The guy on the fringe has the choice to leave the pin in or out. When you're on the green you have the choice to either pull the pin or have it attended.
Etiquette-wise, once everyone is on the green the person closest to the pin usually pulls it (that way the guy with the honor can focus on his putt). In the same fashion, usually the guy that holes out first is responsible for replacing the pin.
When I play we usually play "hit when ready". i.e. the guy in the bunker just skulled his sand shot and now is 40' off the green and is raking the sand. I have a six foot putt, so I putt even though the other guy has the honor.
