There are parks designed specifically to take Fido for some free time to run, services to walk your dog if your schedule is too busy, pet groomers and even psychologists for your pet. This is the world of pets in America but could be in any country in the world.
One thing that is universal is cleaning up after your pet in public areas. Italy has compact cities with limited green space in these cities and not cleaning up after your dog is frowned upon. I have seen in both the US and Italy convenient bag dispensers and trash receptacles specifically for this issue.
So no matter where your travels might take Fido you can always be the responsible owner.