A 'sgambatoio' is a designated, usually fenced, off-leash dog park — the word comes from 'sgambare', roughly to stretch one's legs. They're common in most Italian towns and cities, generally gated for safety, and sometimes equipped with agility obstacles alongside open running space. Signage reads 'area cani' or 'sgambatoio' depending on the region.
Finding one is straightforward: search 'sgambatoio' plus the city name, or check the local municipality's official website, which often maintains a list of registered dog parks along with opening hours and any specific local rules.