coffee enema is a retention enema, so someone doing it shouldn't be lying in their own mess. (Small accidents do happen sometimes, tho.)
Only concern I can think of about doing it on a bed is getting to the bathroom in time when it's time to go. The only suggestion I can give for that is not to play chicken with your colon and to go slow at first with two cups retained for 12-15 minutes each until an entire 12 cups has gone through you. Long process though.
Regardless, the method I use for a coffee enema is to administer it on the floor of a bathroom and then move to the bed because it's easier to elevate my hips there.