The etiquette itself is simple and consistent with good manners anywhere: keep a dog on a short leash, settled beside or under the table rather than up on a chair, and stay mindful of other diners at nearby tables, especially at busy outdoor terraces where space between tables can be tight.
On stone or concrete patios during summer, the ground itself can get uncomfortably hot for paws even in shade — bringing a portable water bowl and something for the dog to lie on makes a real difference. And when in doubt about whether a place welcomes dogs, it's polite — and often expected — to ask before sitting down with a pet: a quick 'il cane può venire?' (can the dog come in?) at the door costs nothing and avoids any awkwardness.