If you weren't at fault for the accident, going through your insurance on your property damage claim will not cause your insurance premiums to go up and will not cause you to be dropped by your carrier. The only downside of going through your insurance company is that you will have to pay your deductible.
Once the other driver's insurance company accepts fault for the accident, they will reimburse you for your deductible. Your insurance company will later go after the other driver's insurance company, or the driver him or herself, and seek reimbursement for the amount your carrier paid on your property damage claim. This is called a "subrogation" claim.
If the accident just happened and there was a police report, you may not even know who the other driver's insurance company is until the police report is ready. If that is the case, and you have collision coverage under your auto insurance policy, you should go through your insurance to fix your car and to rent a car, if you have rental coverage.