If the driver has insurance, that would probably get tapped first, but if they don't have enough it could end up hitting your policy.
When I looked into it for my situation, a one-off thing was fine. You'd get into trouble if you lent your car for an extended period of time or if it was something like you didn't tell the insurance company you had a spouse and they drove the car regularly.
However, basically all insurance in the US extends coverage to people who the owner allows to drive the car.
But if you borrow someones car and you cause damages that go beyond their insurance limits, you can be personally sued for the remainder because you are still liable for your actions as a driver.
There are some exceptions where the owner can also be held liable for damages that someone else does, especially when they do it recklessly (e.g. lending to a drunk driver)
Actually, maybe someone here knows: How much would I expose myself lending my ~$20k car with full coverage to a random idiot? For the sake of argument, say I'm reasonably assured they are legal to drive.