If the other driver has liability insurance, they should pay. If not, the O.D. should pay. If you have collision coverage, use that and your ins. co. will seek reimbursement from the O.D.
Assuming the police came out to the scene, there should be a police report which provides the identifying information for the other driver and insurance information. Go to the police station, obtain a copy of the police report, locate the insurance information and file a claim with the other driver's insurance company.
Your question raises many issues. Did you have a license but not on your person, or no license at all. That will make a huge difference. See an attorney.
Whether or not you or the other driver have licenses is irrelevant to liability. Not having licenses is a matter between you and the State. Liability is a private civil matter between the two of you. The insurance companies should step in and take care of the claims, regardless of whether either of you had a license. Good luck.
Several factors; did the other driver admit liability and did you have comprehensive and collision on your policy? If the other carrier accepts responsibility then they can repair your vehicle. If not then you need to go through your own carrier, but your coverages than become important.
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