I don't have any advice on required sizes as I don't know. There are two separate mechanisms at work killing the green stuff however. First the chickens eat and scratch, killing the grass. They start eating and scratching as soon as they come out of the coop, so the area near the coop will go first. If they have sufficient room, they may not kill everything in their reach, but I don't know what enough is.
The second mechanism to kill grass is that they poop a lot. The poop will build to where nothing can grow. They start pooping as soon as they get out of the coop. (They poop a lot in the coop too.) So the area right around the coop eventually gets a build-up of poop, even if they have a lot of area to range in.
Even if you have no fences to contain them, expect to eventually have some bare areas around the coop.