T.F.
Oh boy.... I am an avid animal lover and I have 3 dogs. If one of my dogs ever bit a child or anyone else...I would be the first one taking responsibility for the pet's actions. I would make sure I covered all medical expenses for the child AND contact Animal control.
I realize accidents do happen and if I were in your shoes, I would base decisions on the severity of the bite. The other family should be more than willing to pay any and all medical costs. It is their responsibility to pay the medical costs.
I believe it is more of an animal control issue vs police. My dogs are registered with the city of Allen and there is a notation on the certificates if there is a history of bites (we do not have bites). It is my understanding that tighter requirements happen to the owners and dogs if they have bitten even once. You do need to know if the dog is up to date on shots. Your child may need a series of the rabies shots and tetanus.
As silly as this sounds....I did hear years ago that each dog has "1 free bite" according to insurance coverages. If the dog is a breed known to bite or attack, then it should be out of the home. Shame on the owners for keeping a dog prone to bite.
What were the kids doing? Sometimes a dog will nip if he is provoked. It sounds like your child got more than a nip and should get medical attention. I'm in no way suggesting your child provoked the dog. Some dogs do "spook" easily and it can happen quickly.
I'm sorry you are going through this. It would be hard for me to go through from either side. As a responsible pet owner and mom, I would be talking with the family to make sure medical costs were covered (their homeowners policy covers this), I would make sure the dog is up to date on shots, and I would contact the local animal control. This bite does need T. on record. The shelter may or may not hold the dog. As much as I love dogs, sometimes they do have T. put down for things such as this.
Best wishes and I hope this your child recovers quickly and well.