It is your own decision to make but if you do not want to pay for your own rental car or pay a deductible then you can wait and place a claim with the at-fault party, if he is insured. If you make a third party claim, instead of a first party claim with your own insurance provider, then you will also not have a claim against your own insurer on your claims history.

If you cannot wait to find out if the at-fault hit and run driver was insured or not then you can make the claim through your own collision coverage. You will need to pay the deductible related to this coverage and will not get repaid by them for any rental car you may need to use during the repair (if your car can be repaired) of your car or during the settlement period.

If you do go through your own insurance carrier then you can ask if they will subrogate with the at-fault party. If they plan to do so then you can ask if they will also try to get back your deductible amount for you.