I think Sweden, yes that "socialist" heaven, introduced an extremely successful scheme of school vouchers. The very basic functioning is that every child receives a voucher annually. This voucher can then be used to go to a publicly run school or a private school. Teachers can become entrepreneurs by buying or renting adequate class room space and then advertising as a new school. These private schools can then be chosen by people of any background, as at present schools cannot accept the vouchers and charge more, but you can make a profit.
The result so far has been that the number of publicly run schools has gone down and the choice available to parents has drastically improved. There are now private schools that teach mainly in foreign languages or that focus more science, or art, or anything that may be in demand. This also weeds out the worst run schools and introduces competition into a state monopoly.
I have talked to several teachers I know, especially those that are not on permanent contracts, about what they would think of the idea of setting up their own school if they could attract the same pupils as a state run school with the same per child funding. So far not one of them thought it wasn't a good idea.