When I was a kid, any of that kind of stuff that was left behind by any of the various horses that came past was gone in seconds and on to somebody's roses.
I can remember some of our neighbours almost coming to blows over some which hadn't even cooled down. They looked at me as if I was mad when I suggested they might split it in two.
You wouldn't do that with the results of a dog going by, it's just no good as a fertiliser. I assume that's because it's the result of meat eating, rather than vegetable matter.