In my experience, all of the onboard dogs that I have been around, go to the bow or "A" mount (or mount 81, 31, 51, or the fwd missile launcher) to do their business.
The crews keep a square bladed shovel nearby to scoop the solids overboard, and a hose to wash everything down.
On a fiberglas boat, I'd recommend a plastic scoop and a 5 gal bucket with a long line tied to the handle. You can use seawater to washdown the designated "pooping spot"!!!
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QMC, USCG (Ret)
The best luck is what you make yourself!