In short, the answer is neither.  Regardless of whether you file a first party or third party claim, you will be stuck between the insurance company who is trying to limit their losses on the settlement and repair facilities trying to maximize their profits.
In many instances, insurance companies have relationships with repair shops in which the shop will offer estimates according to guidelines from the insurance companies in exchange for referrals.  This lowers the cost of repairs but may leave out or restrict some proper repair techniques in order to reduce the cost.  In these situations, the repair shop is working for the insurer’s best interests.
It is best to work with an independent repair shop to get a true estimate for the damages to your vehicle.  In most states, you have the choice of which repair shop you go to.  Your insurance company will offer a referral to their preferred shop, but you are not legally obligated to choose their referral.  When you get an estimate from an unbiased third-party shop, make sure to get a detailed report in writing so it can be presented in the negotiation for your settlement.