Who pays for the damage to my car caused by accidentally hitting a dog running loose in the street?

It depends on fault–not just the dog's owners, but yours as well. To the degree you were at fault (e.g., speeding, texting, running a stop sign, etc.), that would reduce, possibly even eliminate, your ability to recover compensation. If the dog was running loose, its owners are at fault, because it is negligent to not secure your dog. That would argue in favor of them paying. The law balances the relative fault, to see who should pay, and how much.

With very few exceptions, liability—or the legal obligation to pay for damage or injuries—in this country is based on fault: the at-fault party pays. Fault in turn can be based on a deliberate wrongful act, but that’s relatively uncommon: this applies to cases of assault, deliberate vandalism, etc. Most of the time, fault is based on negligence, or being unreasonably careless: the party that acted carelessly, and/or did not take reasonable, common-sense precautions, will be found to have been at fault.

Fault is generally not an either/or proposition: both sides can be at fault to varying degrees. While each state has its own rules for what to do when there is fault on both sides (you will see this referred to as “contributory negligence,” or when the person trying to recover compensation contributed to the accident, or as “comparative negligence,” as in you compare how negligent the two parties are) and you need to reference the law of your specific state, to oversimplify or overgeneralize, if the other side is at fault but you are at fault, too, the extent to which you are at fault will proportionately reduce the compensation you can get. (Example: say the other side is 75% at fault and you are 25%. In that case, you would receive 75% of your costs or damages.)

In a loose-dog case, the dog owner is almost always going to be at fault, for failing to secure his or her dog. (Like with everything else in life, there are exceptions, such as where the dog’s leash or chain broke, or some third party left a gate open.) Therefore, if you were driving carefully, the dog’s owner should be liable for your damage.

But if you were driving carelessly or negligently, your own negligence will reduce what you might be entitled to. For example, say you were speeding, but if you had been going at the legal speed, you might have been able to stop in time. Or even if you could not, the lower speed would translate into lesser damages. In that case, your own negligence (speeding) would reduce what you could recover. There are even cases where your own carelessness equals or exceeds the dog owner’s carelessness, and in those cases, you might recover very little, or even no, compensation. You could even be liable to the owner for the harm to the dog. Examples including texting while driving, so you weren’t even looking where you were going, or driving under the influence or while intoxicated.

To sum it up: if you were driving carefully, typically, the dog’s owner is liable. If you were driving slightly or somewhat carelessly, the dog’s owner will still be liable in most cases, but your carelessness will reduce or offset what you might otherwise have gotten in compensation. But in the event of extreme carelessness on your part, you will have to bear your own repair costs—and may also have to pay for the injuries to, or loss of, the dog as well.