In this case, it may actually be worth consulting with an attorney. Regardless of your own property coverage, the other party is obligated to make you whole. You could have liability-only coverage, but if the other party is at fault, they are required to pay for damages. That said, they will argue what that amount is to make you whole. Your insurance will generally only argue up to what they think is a reasonable number, which may or may not necessarily reflect reality. Beyond that they see no additional return for themselves to get you anything additional.
An attorney (hired by the affected driver, rather than insurance) may be able to successfully argue that the amount of money required to make the affected driver whole is more than what the insurance companies have calculated with their general formulas.
At a very minimum, if I were in this case, I would pay the initial consult fee to at least speak with a property damage lawyer to see if it would be economically feasible to pursue the full loss as opposed to what the insurance companies settle on. The consult fee is usually $50-150, depending upon the attorney. The additional amount recovered may be significantly higher.
In this case, the attorney would take responsibility for negotiations over the affected’s insurance, and the affected’s insurance would probably wash their hands of the matter. Unlike the affected’s insurance, the attorney does in fact have a reason to go for the whole loss amount, rather than just a rudimentarily calculated value.
If the affected had only liability-only insurance, their methods of recourse would either be to negotiate with the other driver’s insurance (and/or the driver itself) for a settlement, or hire an attorney to drive the process. The presence of personal property insurance for the affected typically does not preclude the previous methods of recourse, though once a settlement is reached in any means, other avenues are closed.