The health care system in Germany is not owned by the government. There are hospitals that are owned by the respective town, district or state, but it's no requirement. No one forbids you in Germany to open your own hospital, employ your own physicians and make your own contracts with the health insurers. All practitioners are privately owned businesses, and most of them have contracts with all health insurers. The health insurers themselves are not necessarily governmentally owned either. There are health insurers which in fact are governmentally owned (the Ortskrankenkassen, municipal health insurers). But there are also cooperative health insurers, health insurers owned by private companies for their own employees (called Betriebskrankenkassen, corporate health insurers), and private health insurance companies.
The main difference is that there is a group of health insurers called Gesetzliche Krankenkassen (health insurers according to law), which are heavily regulated and whose service offerings are governmentally controlled. If you earn less than a specified amount of money as an employee, you are required to get coverage from such an institution. Which institution is up to you, it just has to offer you the contract according to law. There are about 100 different health insurers in Germany, which offer coverage according to law, and most of them operate through the whole of Germany. You are free to buy additional insurance if you want more or better services. If you are on social security, you are automatically insured by a Gesetzliche Krankenkasse. If you are self employed, operate your own business or earn more than the limit, you are completely free in your choice of health insurance.