Even if you have paid for the repairs on your vehicle, your insurance company should cover the damage you caused to the other driver if your policy was active at the time of the crash. I would recommend not paying the other party anything until you've contacted your company to start the claims process. It is also possible that you may be reimbursed for your own damage, but you would have to provide proof of the damage before the repairs were made and your collision deductible
will likely be subtracted from any payout you receive. Having as many pictures as possible from the accident scene and all of the paperwork associated with your repair will help your case in getting reimbursed.