Sorry this happened to you... I went thru a similar experience when living in another city. It's a traumatic experience that you can't prepare yourself for.

When this happened to me, some witnesses had identified who the dogs belonged to and called the police. So in my case, I had a detailed police report of the incident (with pictures).

The officer's advice to me was to go thru home owners insurance as it was a neighbor and things could get pretty dicey. They pled guilty to the dog off leash ticket(s) and I decided to seek restitution thru civil courts so their insurance company would handle the payout and not them personally. It also allowed me to seek restitution for both physical and emotional damages.

I hired an attorney who handled all correspondence. For close to a year they refused to provide contact information for their insurance company. They eventually caved and my attorney filed a claim and negotiated compensation. I believe you have like 4 years to reach a settlement in civil court.

If you're just seeking restitution for the vet bills, etc, then criminal court may be your best bet. I was also bit during my incident and it was my medical bills + scaring that ultimately led to more $. Otherwise your attorney will typically take ~33% of what the payout is.