"B&B" has come to be a stand-in for basically anything across the entire spectrum of non-hotel properties - from 100% self-contained units to a bedroom in someone else's house. If someone tells you they stayed at a great B&B, it could have been anything from a private, high-end condo to a tiny attic above a family of 6 (with ONE shared bathroom!). So essentially the level of privacy you get depends on what kind of property you book. Each type of property has its pros and cons.
I have stayed at some beautiful private "B&B"s (condos), and while I love being able to have a nice separate kitchen, dining area, bedroom etc, I miss having staff available to ask questions to or get some quick advice from. If you need help figuring out the dishwasher or wifi, it might be hours before you can get hold of the owner.
I've stayed at some family B&Bs, and while it's lovely having a big beautiful homemade breakfast and people around who can give you excellent insider advice on the area, you do miss the privacy and space of having your own self-contained unit. Also if you're allergy-prone like me, there can be a lot of allergens around - potpourri, cats, etc.
If you're in Italy, I think the best bet is to find what's called an "Aparthotel", but one that has a lobby with staff available. You get the best combination of privacy AND help when you need it.