I would not give papaya. Ever eat a piece? It's like putting a bucket of sugar in your mouth. I have not heard of people who show giving papaya, but I have heard of them giving raisins AT the show to promote poops in case the chin binds up from travel/nerves.
There are better, safer treats than a bunch of sugar. What's wrong with their droppings that you need to give them papaya anyway? Giving sugar to fix a gut issue is kind of like smoking cigarettes to help with a cough. If you are providing a good quality pellet, good quality hay, and a good filtered water, there should be no need to "improve on their droppings."
ETA: Jenn - no, a piece of papaya isn't going to kill a chin, however, most people see "It's okay to give papaya" and start giving it by the handful. They skip right over Susan's comment of moderation (and I still wouldn't give it personally) and go straight to here, have as much as you want.