I think Milton Friedman's idea, was that parents would actually receive education 'vouchers' and be free to use them in whichever education context they chose for their children.
This is slightly different, as the private/public system is wholly handled by the State system: parents simply choose between alternative schools, and the state channels taxpayer's money to whichever school the child attends.
Unfortunately, in many cases this has simply led to corporations or individuals who employ fewer or less qualified teachers, who in turn hand out 'optimistic' grades to the kids (attracting more parents), and extracting enormous profits from the State system.
At the same time, the normal State schools are being starved of resources; standards are falling, leading to more parents putting their kids into the private schools, leading to an inevitable downward spiral.