Half and deseed the papaya, and score some lines into the flesh. Squeeze half a lime over each half of the flesh and add a tiny pinch of salt. Leave the papaya to get to room temperature, and then eat. Trust me on the salt, you only need a tiny amount. It enhances the sweetness.
No, you don't need any lime because papaya is already very delicious but if want to enhance the taste more then you can sprinkle little bit black salt upon it.