If you have recently started your first employment in Germany and you have opted for private health insurance, your employer will most likely confront you with a confusing question: Please pick a public health insurance provider!
But why do you need a public health insurance provider, if you have already signed up for private health insurance?
Simple: the private and public provider have two different functions in this scenario!
The private health insurance provider you have chosen is responsible for covering your medical treatment (depending on the health insurance coverage you have chosen).
The public health provider, however, does not have anything to do with medical treatment in this scenario!
In Germany, they are also responsible for managing the funds of the statutory pension and the unemployments insurance. Basically, they are responsible for withdrawing and managing the share of your income that needs to be paid into these mandatory insurances.
So, if your employer asks you to chose a public insurance provider, they make sure that you are part of the social security system, as legally required for every employee.
To sum up: You do not have to sign up with a public health insurance provider. Instead, you are free to simply tell your employer which public providers you want to use, so they can collect your social security contributions.
Keep in mind that you will not really interact with the public provider, though, since they automatically manage the withdrawal and do not require any intervention from your side. So, there is not much one can do wrong here!