Denmark has had a private school system for decades, and 15% of students go into the private schools (as a parent you pay a portion of the costs so it is not necessarily extremely expensive). "Private" mostly means that you still have to teach specific basics, but you can do it in your way like Montessori.

One advantage of the private schools is they can kick out students they cannot manage easily, while the public schools have to keep everyone. Some recent strutural changes in the public shools are said to have declined their quality, enough so that many government ministers, even of the left wing, are sending their children to the private schools.

I guess it's one of those Nordic myths -- like high minimum wage (there is none in Denmark).