I'm not in Law School yet but I went to private prep and private high school. Will be graduating from a private University (undergrad) but I honestly don't think this makes a difference one way or the other, generally speaking. Just my opinion.
One thing that I do know is for me, the private grade school I attended was British, and I attribute my grammar and knowledge in the English language to the demanding teachers. I also had courses that most people don't have such as British history, Chilean history (the british school I went to was in Chile) among others, but that means nothing in getting into Law School (except for grammar and knowing how to write and read well- always important)