It varies from state to state depending on how private schools are allowed to compete with public schools. In Ohio, public and private schools are all piled into the same conferences and compete against each other. The rules on recruiting students for private schools is pretty lax. So, the catholic schools like St. Ignatious, St. Edwards, etc. load up on the top athletes in the region in basketball and football. The athletes do not get scholarships for sports, but are generally given a free pass on academic standards and usually qualify for need based aid. So, it is a no brainer for a kid whose high school team is not that great to jump ship and go to a private school where they will be in contention for a state championship and get scouted by Div. I colleges.
In other states, there are punitive rules on transfers that keep private schools from stacking their teams. In some states, kids who change to a private school have to sit out a year before they can be eligible for the team. Otherwise, the school cannot participate in the state championships.
In other states, like Texas, the private schools have their own division and do not participate in the state championship with the public schools. Private schools still try to recruit athletes, but the lure of a private school program is not as big, especially if the athlete is already at a school with a strong program.