I'm surprised to hear people talking as if "private schools" were some sort of unified collection of similar institutions.
Some are religious (of all sorts of different varieties); some are non-religious.
Some are small (there is a private elementary school near where I live that has just a few dozen students); some are quite large.
Some will take just about anyone who applies; some are highly selective (like Sidwell Friends School where the President's daughters go).
Some hobble by on a shoe-string budget; some are flush with money (for example, Phillips Exeter Academy has an endowment of ~ $1 Billion and spends ~ $60,000 per year per student.)
While public schools around the country are *relatively* uniform, private schools are so incredibly varied that it make little sense to try to say what a "typical" private school setting is.