is there such a thing as a member funded, non-profit insurance group?

Say a large group of citizens formed an orgnization to equalize risk, say of automotive accidents. Everyone paid a premium to a company that was managed by paid professionals, but say that organization was not allowed to make a profit. Any 'profit' would take the form of reduced insurance premiums.

Does such a thing exist, can you eloborate?
growing inside,can you eloberate?
growing inside,can you eloberate?

What your sort of talking about, is a RECIPROCAL insurance exchange. That is usually for larger scale things, like physician's medical malpractice.

It could also called a "mutual" insurance company, where the policy holders actually own the company.

Any profits left over at the end of the year are redistributed to the policyholders. There are still a few true mutual companies left in the US, but they would vary from state to state.