It sounds as if you later had to call the police though you did not call them to the scene of the accident. It is important if possible to call law enforcement and have them come to the accident scene so that an official report can be made, just in case of things such as this where the at-fault party now cannot be contacted.

Insurance companies have different rules and guidelines when resolving a claim. In general, the at-fault party's insurance company’s claims adjuster cannot contact his insured in a reasonable amount of time they should eventually honor your claim. Presently you should continue to contact the other party's adjuster and discuss your concerns. The should be able to tell you what their rules say will happen if their policyholder continues to ignore them and does not cooperate with the claim.

If you have collision coverage on your vehicle, you may have to go through your own insurance company first and allow them to subrogate to get their money back. You can speak to your agent about this situation and find out if he or she would recommend you place the claim this way or continue to wait for the at-fault party's insurance at this point.

Keep in mind that state laws differ so it might be that if the insurance company cannot contact their policyholder they may be able to deny the claim. For example in Florida the Department of Financial Services, the state's insurance regulator, notes that an insurance company is obligated to pay what an insured is legally liable for because of an accident.

The site goes on to say that a ticket by itself is not evidence of 100% legal liability, therefore, in many instances the company may not have to pay for damages if their insured does not report the accident and cooperate in the investigation. There are many factors taken into consideration when evaluating liability claims. Each situation is judged on its own merit.

Check with your state's insurance regulator to find out what your consumer rights are in this situation. If in your state the at-fault party's insurance carrier does not have to accept your claim if their insured does not respond to their inquires, and you do not want to go through your own insurance carrier (or do not have collision coverage), you may want to seek legal counsel to see if you can take the at-fault party to court on your own to try and get a judgment for your damages.