At any stage, if you are on the green, or even if you are off the green. However, be careful, if your opponents ball is in motion, and you remove the pin without their permission you receive a two shot penalty. Also if you are on the green and you hole a shot with the pin in, or hit the pin, you will receive a two shot penalty.
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No matter what, do as directed by the player. There is a penalty when a ball that started on the green strikes the flagstick. A person tending the pin is considered to be under the control of the player, even if he's a fellow competitor. The penalty would be on the player making the stroke, so avoid making enemies by handling the flagstick carefully.
Some suggestions on good practice:
If the player who is away doesn't want the pin tended then get rid of it. As long as he can see the target, it's better to get the thing out of there, to avoid accidents. Lay it well away from anyone's possible line. I prefer to lay it down on the fringe, to avoid damaging the green.
It is a good practice to get it out of the bottom of the cup before the player hits. Sometimes it can jam in the bottom, maybe even pulling the cup liner up with it. It would be a sick feeling to see a 40 footer curling toward the hole and not being able to get the flag out. Hold it over the center of the cup until after the player hits, then move it out of the way.
Pro caddies like to keep it in the center of the cup as long as possible, probably to give their players the longest possible look at it.