Italy is not really a nation of Animal lovers they are tolerant of them, but despite what people say, they really are not animal lovers. A walk down most streets will show you that, the amount of stray animals and dogs leaving there mess everywhere.
You have to walk your dog on lead, but many people just let them roam free, and there is often a fine in some places of around 30 euros for breaching that, but unless they know its your dog, they have no way or fining you so most people do not care about things like that.
Your dog has to be kept secure, it can be allowed to roam free unless its on your own property.
Vets are expensive in Italy and there isn't as many as you would find in most countries, so pet insurance is normally wise, and most insurance companies offer pet insurance.