Unless I’m misunderstanding it’s no different with private insurance in the US.

If you are injured and the circumstances vaguely resemble any kind of accident (broken bones, concussion, lacerations, etc) a few weeks to months after receiving medical care, you’ll usually receive a letter from the insurer asking for the specific circumstances of the accident including the owner of the property it occurred on and any parties who may have contributed to the accident.

That way they can make a claim against someone else’s insurance/sue them/etc to pass along their costs to someone else if possible.

So telling someone “no worries man if anything happens it’ll be my fault not yours” before getting hurt isn’t going to stop the insurer from pursuing subrogation/lawsuit if they decide it’s cost effective.