Access to specialists mostly need a referral from your family doctor.

That hasn’t been the case in many years now, you can just make an appointment without a referral.

In private insurance, often you yourself will be billed and you will need to hand this to your insurance company.

An experience that’s quite common, as public insurance only covers the absolute bare minimum for dental health (including only the visibly ugly fillings). So many (at least many compared to people having general private insurance) people tend to have additional private dental insurance, and there it works exactly like that, you pay, and then you send the bill to your insurance.

one of the public insurances (there are many)

For those wondering how you decide, it’s mainly about the extras. The one I’m at (TK) is well known for having an English-speaking hotline, which is great if your German isn’t that good, as they also can make appointments for you. One other feature I accidentally found once is that they support OAuth for netdoctor (digital dermatologist visit), while others have to send in forms.

It’s mainly small stuff like that, niche procedures that get covered, paid exercise, and obviously (because it’s Germany) what and how much esoteric bullshit they cover.