What legal rights do you have when you are involved in an auto accident with a young uninsured unlicensed person?

Not sure what you mean by 'legal rights' you are due (assuming other driver is 100% At Fault for the loss), to be put back in the same condition you were prior to the loss, even though they are uninsured and unlicensed, you don't say if the driver is a minor, if so perhaps you could look to the parents for payment. Is the owner of the vehicle different than the driver? If so that is another party to look to. I would file it with my own insurance company and let them handle this. If you do not have collision coverage, then I would start by filing a state report naming all parties.