Do Service Dogs Poop?

Someone I know that has a service dog told me this story a few days ago.

She was getting ready to go into a grocery store with her service dog and she stopped in a grassy area at the side of the parking lot to let her dog go potty before entering the store.  The dog pooped and she picked it up with a plastic bag.  She noticed another customer staring and her and her dog, looking shocked and amazed.

The customer said, "Could I ask you a question?"

The service dog handler said, "Sure."

The customer said, "Do they all do that?"

The service dog handler said, "All do what?"

The customer said, "Poop."

The service dog handler was, naturally, a bit confused.

The customer explained, "I thought service dogs didn't poop.  I thought they bred them that way.  I've always felt bad for them because they can't poop."

Um... how could they breed a dog so it couldn't poop?  But if they do, I want one of those service dogs.  I want a cat that doesn't poop, too.