I want to get a dog or cat. Do pets just rub feces all over your house?

I never had a pet before and I'm sort of a freak about cleanliness. Since pets don't use toilet paper or sanitary wipes, do they just run around with dingleberries? Like if they sit on your couch do they leave behind small bits, microscopic or bigger, of feces everywhere they sit? I don't understand this. Serious question.