The only people who we ever heard say they wanted to be on the Enterprise were the people who were already on the Enterprise, so either they requested the assignment and got it or they were assigned it and then decided that they wanted it.
Dr. Lense and Dr. Bashir were from a class that decided the Lexington was the plum assignment, and Lense got her desired job, but then so did Bashir! So was Lexington really the plum assignment, or was it that all the graduates had their pick of any jobs they wanted?
Makes you wonder how assignments work in Starfleet. If the Enterprise-D, for instance, is so coveted, then how did the likes of Barclay land a position there?