The first thing you should do is to report the incident to your own insurance company immediately. It sometimes takes a while for the other driver's insurance company to complete an initial investigation and contact you. In the meantime, your car may be sitting at a body shop or at a tow yard incurring storage charges. If you had collision coverage on your own policy, your insurance company will act much more quickly to respond to your claim. After all, you are paying good money to your insurance company, so you should use your coverage.
You may be able to settle your property damage claim with your own company. If you had a collision deductible at the time of the accident, your insurance company will subtract the amount of your deductible from the amount paid for repairs. However, your insurance company will seek reimbursement for the payment from the at-fault driver's insurance company, including the amount of your deductible. You should receive a refund of your deductible after your company receives reimbursement from the at-fault driver's company.
If you did not have collision coverage, then you should contact the other driver's insurance company, if that company can be identified. The company will complete an investigation as to who was at fault for the collision. The company will then request to inspect your vehicle to assess the damage and write an initial estimate. You also have the right to obtain an estimate from the body shop of your choice. You do not have to get your vehicle repaired through a body shop chosen by the insurance company (as much as the company would like you to do so).