If your pit bull injures someone, that person's health insurance pays for their care. You might have to pay a few hundred for pain and suffering, right?
Not necessarily. First of all it depends on the nature of the "care", and there could be non-medical expenses as well. Ultimately the owner is financially liable for costs associated with an attack. A personal liability policy would probably cover this.
Interestingly under common law a dog is permitted "one free bite" (sic) meaning the owner is not obliged to keep it under control as a vicious dog until after it actually bites someone.