A lot of times it seems as if talent is hereditary, and some is, but I think musical talent is developed. Now some people were born with good ears, or or "right-brained," but hard work is always needed to improve or perfect musical skill. Before I went to college, I would come up with good melodies, and even whole pieces, but I wasn't good at elaborating on themes, so a lot of my early stuff is "main theme - new theme - another new theme - main theme etc. etc." Theme and variations were completely out of the question. But now I'm going into my junior year and thanks to theory, aural skills, sight singing, composition lessons, instrumental lessons, conducting... and all those other things, those earlier "melodic skills" have turned into "compositional skills." Even though Mozart was composing at the age of 5 he had his father and teacher Leopold, and of course that early stuff is horribly boring!
So in sum, No I don't think that people are born with the ability to compose.