Do you know what papaya variety you have? There's some that will produce hermaphrodite plants, others either male or female. You can't tell which you have until they bloom. Here's a pic of male flowers.

I've also heard that topping the plant can cause a male plant to produce female flowers but don't know how it's done or if it works. I found this online:

One of the most remarkable features of the papaya is the irregularity which it presents in the distribution of the sexes. Normally it is dioecious, with staminate and pistillate (male and female) flowers produced on different plants. Cross-pollination is necessary to enable the pistillate flowers to develop fruits. This is effected by insects. Among seedling plants the number of staminates is usually greater than that of pistillates. Only a few of the former being necessary as pol-linizers (certainly not more than one in ten), this excess of staminates is, from the grower's standpoint, an objectionable feature.

In addition to the staminate and pistillate forms, many intermediates have been observed in which both sexes are combined in one plant. Staminate flowers may occur with rudimentary stigmas and ovaries which give rise to small worthless fruits; and there is a hermaphrodite type which regularly produces perfect flowers, is self-pollinated, and yields excellent fruits. Numerous other forms have been described (see the bulletin by Higgins and Holt), but the importance of these is lessened by the fact that during the lifetime of a plant it may change from one form to another.

In general, it may be said that plants which develop from the seed as pure pistillates will retain their sex without modification, but plants which commence life as pure staminates may undergo a change of sex. It has been asserted that a change of sex may be induced by topping the male tree or breaking its roots. M. J. Iorns, who studied this question in Porto Rico, reached the conclusion that other conditions than the loss of the terminal bud must be present to induce a change of sex, and he suggested that the trees may pass through definitely recurring cycles of development, and be subject to the change only at certain periods. L. B. Kulkarni, 1 who investigated the matter in India, came to the belief that change of sex is not in any way connected with the removal of the terminal bud. He found that male plants, in the course of their development, may present a number of different sex-combinations, as follows:

First stage: Staminate flowers only. Second: Staminate, with a few hermaphrodite flowers. Third: A few staminate, with many hermaphrodite flowers. Fourth: A few staminate, with many hermaphrodite, and a few pistillate flowers.

Fifth: Hermaphrodite flowers only. Sixth: Hermaphrodite, with a few pistillate flowers. Seventh: A few hermaphrodite, with many pistillate flowers. Eighth: Pistillate flowers only.

Thus the plant in the course of its life history may change from a staminate to a hermaphrodite and then to a pure pistillate.