Well, I know you don't 'know' me, but I can tell you my experience with private vs. public
I was offered out of the blue a position to head up a newly formed learning disabilities department at a private school in our area. Tuition at this school runs between 9-12k a year (depending on grade). I was told that one of the advantages was that I would no longer need to maintain my certification and therefore save hundreds of dollars a year in extra classes. I asked why and was told that in the Commonwealth of Va, private school teachers did not need to adhere to certification standards. The salary offered to me was less than 1/2 of what I make in the public schools. Why? Well, because the private school doesn't have to have state qualified teachers. Two of the teachers at that school do not even have college degrees. Granted, these teachers do not teach core areas - PE and art. But they come cheap.
That is my experience, others in other states may have a different one.
BTW, there was an article in the WaPo recently discussing the salaries of private school principals and 'heads.' These people earn much more than a superintendent of a public school system. Typical salaries ranged from 300k to 500k with huge perks. Now we know where the tuition money is going.