I studied twice at the Universitą di Pisa summer school for foreigners many years ago. It was then held in Viareggio in a boys. boarding school. Some students lived in the school (I was one) and others stayed with families in town. Several years ago, I attended the excellent Koinč School in Forence. They have schools also in Bologna, Lucca, on Elba, and in Cortona. I stayed in a nearby convent, which I arranged myself. In three weeks I leave for Taormina (Sicily) where I'll attend the Scuola Babilonia for a month. The school is well-regarded and several people I know who have been there say it's excellent. I'll be renting a studio apartment, which the school arranged for me.
Most language schools will help you arrange accommodation. You don't have to go through an agency. I know there are agencies on the Internet that will arrange the whole thing for you -- school and accommodation -- but you pay a hefty price because they mark up everything. Cheaper to do it through the school. They usually offer independent or shared apartments, home stays with families, or can suggest local hotels and pensioni. I know several people who have done home stays. Reviews are mixed. If you get a family or a person who is interested in knowing you and socializing enought to support your language learning, it's great. However, some hosts apparently just want a little extra income and make little effort to include their student guests, so it isn't a friendly experience. A couple of people I know had disappointing home stay experiences -- then again, others enjoyed their home stay hosts.
Whatever accommodation you choose, you'll meet people to go around with once you're at the school. Don't worry about being alone for dinner!