As a teacher working in one, I feel qualified to answer

All schools, private or public, are required to follow the same basic curriculum, but with some space for specialisation. Education is funded by the local "kommun" (local government, population mostly somewhere between 5000 and 200 000), and a private school gets paid as much for each student as the kommun has costs for a student in a public school. Outside of this, private schools are free to seek sponsorships from companies, but anyone wanting public money cannot also charge fees for attendance. One of the discussions about whether this is fair is because public schools have to a) accept any students and b) pay for administrative overhead that private ones don't (think a local department of education).

Anyone must be allowed into a private school, allowed criteria to select students are queue time, closeness and sibling ties. If there's a focus on music or dance, tests in this are allowed, not so for other subjects. After year 9, grades are used just as they are for public secondary schools.

Private schools can be run and owned by associations or companies (including for-profits, which is a sticking point for the left). In some cases this has led to large corporations who run schools in many Swedish cities, while the one I work at is owned by the family who were involved in starting it.

All schools have to be certified by the School Inspection, and they are currently tightening the rules so that they'll be closer to what the public schools face.

Talking more about my own case, we are a secondary school (year 10-12) that has a focus on horses, with our students focusing on either jumping, dressage or harness racing. We offer theoretical profiles (natural or social sciences) as well as an option more focused finding a job straight after school. It's only our second year, and we have around 15 students in each year (after setting our limits to 20, but neither year filled up).