It’s not complicated, but there are a few moving parts.
Person enters into a contract with Shifting Sands Mutual.
At minimum, the contract (policy) will provide coverage for bodily injury and property damage caused when an insured driver causes a covered, at-fault accident. The amount the policy will pay for such things is determined by the limits the Person chose when they bought the policy.
In addition to paying up to the limit of the coverage, the insurance company will also pay the attorney and court costs to defend the Person in the event of litigation.
If Bodily Injury claim exceeds the coverage limit, the insurance company does NOT just write the check and close the matter, leaving the Person to the tender mercies of the damaged party. Why? Because the only way for the damaged party to effectively get money out of Person is to sue them–which the insurer will have to pay to defend. Therefore, the damaged party signs a release of all claims against the Person, in exchange for the maximum payable under the policy. Bang, Person is now immune to being sued by the damaged party. It DOES sometimes happen that the damaged party elects to not settle, and instead proceeds with litigation. Unless the Person actually has large reserves of assets, however, this is usually a fool’s errand. There are exceptions to the above, but those are exceptions.
Similarly with property damage claims. The way that usually goes is, Person buys $10,000 in property damage coverage and then goes out and plows a Corvette. Not enough coverage. Corvette owner has three options. 1) Sue the Person and get the money (less the $10k available from Shifting Sands Mutual), which they will then have to collect (good luck with that); 2) take the $10k and be sad they weren’t fully compensated for their Corvette; 3) seek recovery from their own Collision coverage from their policy with Furthest Farthing Insurance, in which case they have to pay a deductible. So since the OP is asking about option 3, let’s see how that works.
Collision coverage pays you up to the actual cash value (what you’d get if you were to sell the car today, WITHOUT the collision damage) less your $500 (or whatever) deductible. So, for your $60,000 car, Furthest Farthing will pay $59,500 and take the car. Your insurance company doesn’t like being out that $59,500 so, thanks to the subrogation clause in your policy (I would be stunned to see a company that did not preserve its right to recover a loss) Furthest Farthing will go about looking for the money. If the at-fault party his minimal assets (they can run an asset search through public databases), they will take Shifting Sands’ $10,000 and be sad. But if the at-fault person has a hoard of sufficient volume of UNPROTECTED ASSETS, they might just go after them personally. Litigation is expensive, however, and that cost is considered when deciding whether to get happy, or just remain sad.
But at bottom, your insurance company will not just write a check for your policy max and bid you good day. Their covenant is to protect you as much as possible from what you’ve done. That means fighting as hard as they can to minimize your personal accountabilitiy for the damage you caused.