I free range my geese during the day, and I put out plastic totes with water they can drink from and bathe in (among other things). In their coop at night, I have a 3 gallon top-fill waterer, and a 7-lb feeder. 8 geese (about 3 months old now) don't really make a mess at night with the water. The drink only what they need to eat the feed.
I have another coop (more of a large chicken tractor) with muscovy ducks and geese (mostly ducks) and they're 1.5 months. The ducks splash every ounce of water out of the waterers (I mean, I'm assuming it's the ducks, I don't stand there and watch them). They make quite the mess.
Based on my limited experience, if you only have geese in the coop and they have access to bathing water during the day, they probably won't make too much of a mess. If there are also ducks in the coop, big mess.