UM payout - will USAA go after at fault driver?

I was recently read ended which resulted in the total loss of both cars. The other driver was deemed at fault but I went through my own insurance, USAA, for the property damage and PIP. I live in FL which is no fault. Apparently, the driver did not carry bodily injury coverage, so my medical claim is being handled by a combination of my own PIP and uninsured motorist policy.

I have settled these claims with USAA. My question is, do they typically go after the at-fault driver after the fact when they have to payout for UM or do they just eat the cost?