I have a large, long haired mutt. I try to take her ashore every morning and evening. If I can't, she'll give me about 24 hours before going on the deck. She usually pees along the rail, and when I find it, I get a bucket and rinse it off with sea water. She usually poops on the cabin roof. That gets picked up with a poop bag, and stored in a gallon zip lock bag on the aft rear deck until I can dispose of it properly. It usually comes up cleanly, but if not, the deck gets washed with sea water. Even in a zip lock bag, it will stink up any other container it is put in, and my holding tanks are too small to use up flushing dog stuff that has already been bagged, plus I'd still have to deal with the dirty bag.
It is no more legal to dump dog waste in the water than it is to dump holding tanks - you should be 3 miles off-shore before you do.
I've tried pads, but its easier to clean the deck than the pads.
When we go ashore in areas without trash facilities, the poop also gets picked up, taken back to the boat and put in the zip lock bag to await proper disposal.