Oddly enough, private schools can pay a fair bit less. At a private school, you get to avoid most of the disruption, have smaller class sizes, and teach brighter students on average. If you're passionate about what you're teaching, these are things you may value more than salary, so...
Private schools also have fewer hiring restrictions, which lets them lower salaries. I know a couple people who did not get the qualifications to work at a public school, and now teach at private schools. (Not for lack of ability, by the way).
Really? Our state generally just requires a state teaching certification and FBI background check. I thought that was also a requirement for K-12 private schools.