Dogs can safely eat the Papaya's itself but not the Papaya seeds that are contained in the Papaya.
The Papaya with the seeds removed is a healthy treat for your dog that they can safely eat.
The Papaya seeds should not be given to your dog because when the dog ingests the Papaya seeds the seeds from the Papaya can cause intestinal blockage and the Papaya seeds also contain Cyanide which is harmful to dogs.
So like I said you can give the dog the Papaya as long as the Papaya seeds are removed before giving the Papaya fruit to your dog.
Also cut the Papaya fruit up into small pieces before giving it to your dog to avoid choking.