Understanding there are about 51 different sets of rules in the USA alone, and about a gazillion subtle details that can dramatically change the nature of a case:
Insurance follows the car: ordinarily true, however there are a few companies that will not provide coverage if the driver is not the named insured. But usually, this phrase means that if The Jester is driving Nars’ car and causes an accident, Nars’ policy will be primary and The Jester’s policy will be excess (kick in only when Nars’ policy is insufficient). However, most policies also provide coverage for Non-Owned Cars. Which means The Jester could, If he wanted, file the claim for the accident under his own insurance, making IT primary, and leaving Nars’ as excess. This keeps Nars happy, because he doesn’t take the premium hit for something The Jester did, and it keeps The Jester protected in case Nars is a cheapskate and has low liability limits ( )
Injuries to other people: This is called a Bodily Injury claim. With a few exceptions (Montana being one) a BI claim does NOT handle bills as they are incurred, but is instead paid as a lump sum settlement in exchange for a release (a document that says, in effect, “Nobody was liable for this accident, but if Electronbee will agree to not pursue The Jester and Nars for bodily injury damages, Shifting Sands Mutual will give Electronbee $23, 420.”
OP’s Question was who is financially responsible?: The driver of your car is the one who caused the accident and is primarily responsible. If you, the car owner, know The Jester to be a drunkard with a reputation for driving furiously, and he reeks of alcohol because he’s loaded as a freight train and everyone around you knew it when you handed him the keys–YOU also become responsible for the resulting havoc through the doctrine of “negligent entrustment.” That means just what it sounds like it means. Also, if the person you loaned the car to was driving it “on the clock” for work purposes (Pizza guy, window salesman, courier, etc.) then the company he works for would be “vicariously liable” for any accidents such a driver might cause.
Attorney settlement demands: Almost every time, an attorney will begin BI claim negotiations by requesting/demanding the policy limit. The initial demand means nothing. The insurance company SHOULD counter with an opening offer that represents a possible jury verdict based on the known facts, and not less. The goal of the insurer is NOT to settle the claim for as little as possible, its goal IS to protect their insured, which means attempting to settle the case for a reasonable amount without exposing their insured to the uncertainty of a jury trial. Which is not to say trial is a bad thing–In my experience about 90% of jury verdicts are much lower than my best prelitigation offer.
Claims worth more than the available insurance & assets: Collectability (income attachments, forced property sales, etc.) varies wildly from state to state, and from situation to situation. Injured parties need to make a choice: take the money that’s on the table, or go for a verdict which may or may not be collectable.