Fascinating. In my country private schools will get almost the same amount from the government per pupil as public schools, provided they keep their fees at a reasonable level and comply with certain quality standards. Same with private hospitals, btw. They will get the same amount per patient as it would cost to heal them in a public hospital. There is real competition between the private and public sectors in areas where the population is dense enough, for specialized services like heart surgery, and for boarding schools. If the private sector can provide a better service at a lower cost, the private sector will win. For some reason the private alternatives almost always cost more. I.e. the sum of user payment and public refund is higher than the cost of the same in the public sector.
There seems to be a mistake in terms. If your "private" schools are getting government funding, they're not private schools. They're public schools that also charge their students. You see "real competition" between the "public" sector and the "private" sector, when, in fact, you don't
have a private sector.
Yes, there are some schools which are run for profit, and don't want to keep their fees below the limits for public funding, or wants to have very different standards (not fulfilling all of the minimums, e.g. so their pupils will have to go an extra year to qualify for higher education). A couple of those are popular among dropouts who want to pull themselves together and study intensively to qualify for higher education.