The state always gets paid first. Most private contractors who perform this service won't take cases where state benefits exceed a certain amount. Otherwise, they would be working for the state.

They do get paid a percentage of recovery based on fees that the non-custodial parent will pay at the time of garnishment, lien, or recovery, and the custodial parent will have to take a hit in what they would have received in total.

Better than what they would have gotten otherwise.

The best scenario would be a balance of private contractors and state enforcement for recovery of back child support. State enforcement working on recovering what is owed to the state to recoup expenditures in benefits paid. Private contractors working on all other claims. Of course, the custodial parent would have the option of choosing since going the private route would cost them something.