The decision will really be up to you. If they want to pay out of pocket I would take it to a shop you want and trust and get the estimate. If they want to pay it that's great. If they are unwilling to do that, then you can involve insurance companies. If they refuse to give you their information, you can contact your insurance company. They might recommend filling a police report. Although the damage wouldn't be covered through their car insurance, it would be handled through their homeowners
or liability coverage. Your last resort would be to use youruninsured motorist property damage coverage
. While this comes with deductible, your insurance company might reimburse you for it and go after the at-fault party. Best of luck.