Those that are church-related may receive funds from the church. Church schools are funded by tuition, gifts and the church. Some of the private schools get money from public school systems, out of monies public systems receive and must share with all schools in a locality. Some, especially the independent schools, have endowments. Most of the schools have Boards of Trustees who guide the school's direction, but the school administration they hire runs the school, hires teachers, etc.
Different states have different regulations for private schools.
You'll need to check with state departments of education about this. Most of the high schools seek certification from the Association of Colleges and Schools. Some of the private elementaries have this certification also. I have searched the internet and got 729 matches. Early Childhood Education at the University of Illinois.