They do but not to the extent of the private ones. The public schools are at a disadvantage because they can't offer financial aid like the private schools can, as far as i know. There is also now a rule discouraging OIA transfers too. Public schools are more restricted to having players living in their districts. more than the private ones.
There are grumblings about Bishop Gorman in Nevada too.
http://thegrizzlygrowler.com/2016/10/24/unfair-advantage-should-eliminate-bishop-gorman-from-d4a-competition/