This is essentially free market gun control. Insurance companies would determine the best way to evaluate gun owners. They would set their actuaries to work to research facts and statistics in order to set appropriate premiums, or in high-risk cases even to withhold a policy. They would require mental health records, and would adjust the premium according to the owner's training, family and work circumstances, type of gun, and so on.
Insurance policies would pay for damage caused by the use or abuse of a gun.
With an insurance requirement, the price of owning a gun would reflect not only its market value but also its impact on society, much the way we already do with cars.