Either can be greasy, or not, depending on how they’re cooked. We have duck because a goose would be far too big when there’s only three of us at home.
Cooking tip (for duck – I have never cooked goose so I don;t know if it would work): score the skin deeply all over to make channels for the fat to run out. Peel a load of potatoes, cut them in half and layer them, cut side down in the bottom of the roasting pan. The potatoes lift the bird up off the bottom of the pan so it’s not sitting there frying in its own grease. You’ll still have to drain it off several times during cooking, the important thing is to make sure the bottom of the bird is always above the drained grease, but potatoes work better than a rack because they actively draw the fat out. Throw the potatoes away after cooking, they won’t be edible.
We use stuffing in the bird because stuffing in a duck goes far too greasy and horrible, but instead, put a whole, unpeeled orange in the cavity. The orange flavours permeate the meat and it’s wonderful.
This is exactly what I’m having for Christmas dinner.