I myself am a pet lover, I have owned dogs and cats,
I looked up the meaning of the word pet in the dictionary here is what it means. "an animal kept for amusment or companionship"

1--So if your cat or dog craps in you yard, you may find it a amusing but when they do there business in my yard that to me is not amusing

2-- the word companionship in the dictionary means "to accompany" ( or to go with). so one could say that an animal that is not in the company of ones owner may perhaps be considered " Not their Pet "

Hope this helps a little.