Those injured usually have the law on their side. In the old days, the law gave dog owners what was called "one free bite." Put simply, an owner wasn't liable for injuries unless the dog had already shown it was likely to hurt someone. The dog didn't actually have to have bitten someone -- for example, if your dog lunged at the neighbor, teeth bared, you were considered to be on notice that the dog might bite.
But most states now make owners liable for any harm their dog causes, whether or not the owner had reason to suspect that the dog was dangerous. Dog owners can find themselves on the hook for an injured person's medical expenses and lost wages, or even the therapy bills of a traumatized child. The dog owner may not be liable if the dog was provoked, or if the injured person was trespassing, but claims like these are often very hard to prove after the fact.
It's far better, of course, to avoid injuries rather than fight about legal liability after they happen. And the truth is that dog owners could prevent most bites.