This can be a tricky situation. Usually, someone will try to avoid insurance companies for two reasons; they either don't have any insurance or they want to avoid their rates going up. If the damage is small, this is generally an encouraged idea. You should simply go to a mechanic shop you trust and get the estimate for the damage and have a check made out to you. Keep in contact with the at-fault driver in case anything else comes up and you need additional funds. However, you are under no obligation to do this. If you don't trust the other driver, get their car insurance information and contact their company and begin the claim's process. If you're looking for more resources regarding how to file a claim and the process you can expect, see our guide here.