No, you should not try to get paid by both insurance companies for the same claim since this would be likely be looked at as a form of insurance fraud. With one of the insurance companies paying for the damages to your vehicle you are made whole and so you should not try to collect twice and put yourself in a better position, it is typically against the law in fact to recover twice for the same incident in this manner you are describing.
Since you were not at fault in the accident you normally have two choices of how to make a claim for the damages to your vehicle. One is to file a claim under your own Collision coverage. If you do make a first party claim with your own insurance company you do have to pay the deductible amount and unless you have coverage for rental car reimbursement you will have to pay for a rental car (if one is needed while your car is in the shop being fixed) out of your own pocket.
Many people instead take the second choice which is to make a claim under the Property Damage Liability coverage of the at-fault party. When you make your claim this way you do not owe a deductible and usually a rental car will be paid for by their insurer while your car is being repaired.
So you can go through the at-fault party's insurance, which it sounds as if you have, and have them pay for repairs however you would then need to tell your insurer that you received payment for your damages from the other party's insurer so that your Collision claim was closed out with your own insurer.
If you instead took the money from the other insurer and continued with your claim under your own Collision coverage and have your car fixed that way and tried to pocket the money from the third party claim over the deductible amount you would be found out by your own insurer. When you file a claim with your own insurer and you are not at fault in an accident they will settle with you and go back after the at-fault party's insurance company for repayment, this is referred to as subrogation.
In your car insurance policy there is language that automatically gives your company the right to pursue anyone else for repayment (the process of subrogation). Since you have already been paid by the at-fault party's insurance company they will provide your auto insurer with proof that they paid you. Then your company will come back after you to get repaid since they will now realize you have been paid twice for the same incident. They can and will likely sue you if you do not return the money.
So if you do not want to pay the deductible, or cannot afford to do so, go through the at-fault party's insurance company to get your car fixed. However unless you want to have difficulties and possibly a claim of insurance fraud placed against you, you should not continue with the claim with your own insurance company for those same damages and try to get paid twice.