What can I do legally when both car insurers claim their insured drivers are not liable in a car accident?
An insurer's determination of fault is not legally binding--only courts can issue legally binding determinations. Therefore, if neither insurer takes responsibility, you may sue the driver(s) you believe to be at fault and let a court determine if you are entitled to compensation.
Ideally, when you are in an accident, all the parties involved (drivers and insurers) will agree as to who was at fault, and the at-fault driver's insurer will pay compensation to the not-at-fault driver. However, the real world is not always (or even necessarily often) ideal, and sometimes it happens that neither insurer will admit or agree that their driver is at fault. When that happens, what can someone who believes they are owed compensation do?
Fortunately, there is an entity whose entire purpose is to decide who is at fault and award compensation: the court system. Therefore, your recourse if the insurers will not voluntarily offer compensation is to sue the driver you believe to be at fault.
Insurers may be large, bureaucratic organizations with deep pockets and lots of expertise, but legally, they are actually no different than the drivers they insure in a critical way: similar to their drivers or policyholders, they are essentially private "citizens" who do not have any legal, judicial, or governmental authority.
They cannot determine fault in a legally binding and enforceable way. They can (and generally do) have an opinion as to who is at fault, and they try to convince the other parties to agree with them, or may unilaterally offer or propose a settlement which the other side may choose to take, but they can't enforce their opinion. Their opinion as to fault is no more binding than the opinions of drivers in the accident.
Often, everyone will more or less agree and the matter can be settled voluntarily. In a high percentage of cases, the facts of the accident make clear who was at fault (e.g., a car stopped at a light is hit from behind; the rear driver is at fault for hitting a stationary vehicle). But there are ambiguous cases as, for example:
Car 1 "T-bones" Car 2 as 2 went through an intersection controlled by a traffic light. Who was at fault depends on who had the light.
(1) if car 1 had it, then car 2 must have gone through a red to get into the intersection, into position to be hit;
(2) if car 2 had the light, then car 1 must have been running a red light and hit car 2;
(3) If there were no cameras at the intersection and no witnesses other than the drivers have been located, each insurer will typically rely on what its own driver says.
(4) If the two drivers each insist that they had the light in their favor, the insurers will usually be unable to come to an agreement as to fault to resolve the matter, resulting in a deadlock.
Court cases, however, cannot be deadlocked. If one driver sues the other, the court must find either that the plaintiff (person suing) proved that defendant (the person being sued) was at fault and award him compensation. Or else find that the plaintiff failed to prove in his case and therefore deny him compensation. (This is slight oversimplification: in many or most states, you can be partially at fault, with the degree of your fault influencing how much you will be paid if you are the plaintiff, or have to pay if you are the defendant. However, the principal is the same--the court makes a legally enforceable determination or finding as to fault.)
Courts cannot leave matters unresolved. They must find fault, and therefore liability, which means they have to determine who has to pay how much to whom. After a court determines liability, then the at-fault driver's insurer must pay the judgment, at least up to the policy limit and in accordance with the terms of the insurance policy.
Therefore, if the insurers each deny fault on the part of their drivers, the way to resolve the case is to sue the driver whom you believe to be at fault for the damage he or she caused.