UM/UIM coverage (uninsured/underinsured motorist coverage) can be purchased in addition to liability insurance, and protects victims when drivers do not have auto insurance or do not have enough insurance to cover losses.
A car accident passenger might have UM/UIM coverage, through which they can obtain compensation for medical expenses over the at-fault driver's limits. UM/UIM may also cover a hit-and-run accident. Additionally, your health insurance may cover medical costs if you are injured as a passenger.
In the event of a successful lawsuit against the at-fault driver, your health insurance company may legally demand repayment. This is called "subrogation of a claim."
A vehicle insurance company could sue the insurance company of the other driver in the event you fail to file a claim against the at-fault driver. Depending on the amount of expenses paid on your behalf, your insurer would decide whether it is worthwhile to seek reimbursement from the at-fault driver's insurer.