Responsibility to what? You have to think there is a problem with dog shit in grass to think you have the responsibility to clean it up. I don't have a problem with it. Infact, it's a pretty normal phenomenon to come across (at least around here) and people really think nothing of it.
Responsibility to pick up the mess your dog makes which you've agreed to take care of? I imagine if the dog shit in your house you would clean it up, or would you leave it too? lol
And I don't have a problem with shitting on people's front door steps

See where this is going? It's all a matter of opinion so using excuses like the ones you used just don't fly when trying to make your point to the other side...it's like "oh it's ok because I think it's ok".