Embassies and consulates in Italy
The first thing to know: As a foreigner abroad you will always deal with your country's local consulate, never your country's embassy.
Unless you are a diplomat, on-duty military, or get invited to a really fancy party, there is no reason to (and likely no way into) your home embassy. Embassies are where your country's ambassador and his or her staff do their jobs.
What you may be looking for is your home country's local consulate, which is the office that can help out foreign visitors to Italy with services and needs—such as lost passports.
Every country has at least a main consulate in Rome, and many have consulates in other major Italian cities as well. (For the record, the U.S. also has consulates in Florence, Milan, and Naples; Canada, the U.K., and South Africa also have consulates in Milan.)
Note that many consulates require appointments—and even if not, an appointment will make the interminable bureaucratic wait much shorter.
Yes, in a given capital city, your home consulate will likely in an annex of your home embassy. However, the consulate is maintained as a separate section (an ID and an appointment—or loooong wait—will get you into your home consulate; burly Marines will keep you from ever getting into the embassy), and while there is only one embassy there are often multiple consulate, one in each major city.
Also, security tends to be, understandably, strict. Leave at your hotel all bags, backpacks, oversized purses, knives, glass bottles, laptops, cameras, and anything else that would raise eyebrows in an airport security–type situation. (Many also insist you keep your cellphones in your pocket or purse when on the premises—basically so you can't take pictures, which is a security risk.)