Your question is clarified in the comments as
Looking for a breed of dog that will walk himself to the park and drop his feces under the bushes.
And the answer is Kind of...
Given sufficient freedom and room to roam, almost any breed of dog will do his business out of sight, and where neither him nor you are likely to step on it.
I used to live out in the country, on several acres surrounded by forest. I had a dog most of that time. For years I never had to scoop poop unless the dog was restricted to a small area for a long time. In fact once I had some of the acres fenced. I never even saw dog poop, they went out into the woods to do their business (go poop).
Your problem is legal. The term "Leash Law" applies here, it applies mostly to dogs, there are several posts on the internet about it but in short.
- If there is a law, you have to keep your dog leashed, fenced or under direct control at all times
- If there is not a law, and the area is developed your dog is subject to being impounded (taken to the dog pound) if it is not under direct control.
- In the absence of laws or risks of impoundment, you are liable for any damage to people or property your dog does.
Theoretically the same laws apply to cats, in practice cats are at less risk of having these laws applied.