When you refer to private schools and increasing your chances of specialization, I'm assuming you mean schools like Columbia, Harvard, Penn. From my experience, all my classmates are very open with each other about what they want to do and everyone supports each other. There isn't this cut-throat competitive environment that I've heard some of my friends at state schools face. Having a class ranking truly brings the worst out of some people unfortunately.
As for the personality of students at private schools, it depends on your class and not necessarily the school. Each class is different and has their own personality. Just like every dental school, you're going to have some people who have their education paid for by their parents, some people who are on HPSP, and some people who take out 100% loans.