Based on random videos I have seen, it seems pretty difficult to shot down a drone with a shotgun. Sure, better than nothing, probably your best bet at that point.
Clay pigeon / skeet shooting is hard too, it's a competitive sport for a reason, but then, so is shooting a target with a rifle at 400 yards, but the armed forces train you up on that. So designated drone marksman might become an interesting new speciality in the modern army.
