perhaps not many Danes go private is because they invest so much a country in state (taken from above TES article) :
DENMARK'S SYSTEM AT A GLANCE
Early years: Compulsory education does not start until age seven, but Danes enjoy unrivalled access to state-funded, pre-school education and childcare. Parents can send their offspring to day nurseries from the age of six months until they are three and to kindergarten from three until six.
Primary and lower secondary: Although formal education does not begin until seven, 98 per cent of Danish children start at six, spending their first year in a pre-school class. They then do nine years of compulsory education, usually keeping the same class teacher through the primary years and often beyond. Most attend public schools or folkeskoler, run by the local authority, though one in eight attend publicly-funded private schools, called friskoler. Pupils can leave at 16, but can stay on for a year of specialised study.
Examinations and testing: Testing in Danish schools is common but is used to check pupils' progress, and results are rarely published. Most 16-year-olds take a school-leaving exam at the end of the 9th grade (in Danish, maths and English, physics/chemistry, plus German or French) with the option of sitting an advanced leaving exam in the 10th grade. Sitting the exam is optional, but in future schools will publish results.
Class sizes and conditions: The average number of pupils per class is 18 in local authority schools and 15 in private schools. Pupil-teacher ratios are even lower, with local authority schools enjoying ratios of 10:1 and private schools 11:1. A typical teacher works a 42-hour week during term-time and works for 200 days a year. This is likely to include teaching 23 lessons a week, 40 weeks a year.
Post-compulsory education: More than nine out of 10 pupils stay in education or training after 16. Sixty per cent take job-related courses, lasting between one and four years at privately-run commercial and technical colleges, which are state-funded. One in three pursues academic or general vocational courses, lasting two or three years, in public or private gymnasiums (upper secondary school) or in specialist business or technology colleges. Most of these (70 per cent) go on to higher education.
Cost: Denmark invests more public funds in education than any other country in the world. Comparative figures (OECD, 1998) show that it spent 6.8 per cent of its gross domestic product on education compared with Britain (4.7 per cent) and an international average for developed countries of 5 per cent.