and here's the photo of the 4-month-old tree...

Birds can carry the seed a long way so it's not surprising to see them spring up anywhere. The second photo looks like a male tree. However, you don't need a male in close proximity to get pollination. Bees are very effecient at carrying the pollen long distances.

Thank you, tropicbreezent. Are you fairly certain that the second tree is male? I ask because I was just given two potted papaya plants that purportedly came from Cuban seeds, and though I really don't have the space, I would plant one if it turns out that I have two female trees.

Just as you've mentioned, I have read that bees can pollinate a female tree, but it doesn't seem to be happening with mine.

Males tend to flower more prolifically and the flowers are often in dense clusters. And ovaries in female flowers are usually very obvious as in your first photo. I'm in a rural area so gardens are far apart. There's no other plants around I can see that might pollinate females at my place but they've always ended up pollinated when I haven't had any males.

Thanks. Do you fertilize? I'm trying to figure out why the fruits don't seem to grow any bigger than the size you can see in the first photo.

They like alkaline soil, high pH. Similar to passionfruit, around 8, whereas most other plants prefer around 6.5 to 7. Generally I don't fertilise other than to sprinkle some wood ash around them. I tend to leave them to their own devices. But you could try something with a higher potassium content and lower nitrogen. Potassium is good for flowering and fruiting. Nitrogen is good for growth and leaves, but too much can cause fruit drop.