(4) The right to establish private schools shall be guaranteed. Private schools that serve as alternatives to state schools shall require the approval of the state and shall be subject to the laws of the Länder. Such approval shall be given when private schools are not inferior to the state schools in terms of their educational aims, their facilities or the professional training of their teaching staff and when segregation of pupils according to the means of their parents will not be encouraged thereby. Approval shall be withheld if the economic and legal position of the teaching staff is not adequately assured.
(5) A private elementary school shall be approved only if the education authority finds that it serves a special educational interest or if, on the application of parents or guardians, it is to be established as a denominational or interdenominational school or as a school based on a particular philosophy and no state elementary school of that type exists in the municipality.
(Emphasis mine). Private schools over here are generally either confessional, follow different pedagogic approaches (e.g. Waldorf, Sudbury) or, last but not least, serve a national minority, e.g. there’s plenty of Danish schools in northern Schleswig-Holstein which are, legally, private schools but teach to the Danish curriculum (in Danish) while making sure that kids also get German graduation papers. And yes they generally all receive state funds. Can’t find proper numbers right now but ballpark 75 to 85% of what public schools get per student.