Italy and China have about the same number of reported cases, but Italy has more than double the number of reported deaths. Does this mean that there is something special about the Italian epidemic (e.g. older people on average, or a collapse in the healthcare system that means many people are dying who could otherwise be saved), or could it be that the Italian epidemic is simply much larger than the epidemic in China, with many more unreported cases? A serological survey (testing blood for antibodies to the virus produced by white cells) gives us a handle on this. If this is a large, recent, representative sample of the Italian population and the results are correct, then it does imply that large numbers of Italians have been infected already, perhaps tens of millions rather than the tens of thousands we currently know about. This could explain why there are so many deaths, and in fact the proportion of deaths begins to look relatively low rather than relatively high.
The antibody test wouldn't tell us how many are currently infected, just that they've been exposed at some point, though because some types of antibody (IgM) are produced earlier than others (IgG), it may be possible to draw inferences about what stage an individual's antibody response is at if the test detects both. To detect a live infection you need to swab the respiratory system and do an RT-PCR test that picks up the genetic material of the virus (all the testing you've heard about up till now will have been done this way - it's the easiest test to develop rapidly). There'll be a period when a patient with an active infection is positive both by RT-PCR and by the antibody blood test, but as the infection is cleared they will only be antibody positive (hopefully for a long time). You could therefore use antibody testing to identify immune individuals who could (say) return to work safely.
Edit: Coronaviruses aren't thought to be bloodborne or to persist long-term, so it's not like being HIV antibody positive (which implies your body fluids are permanently infectious unless the virus is suppressed by medication). I believe viral RNA is, in a proportion of cases, detectable in the blood by RT-PCR, but I don't think anyone has found infectious virus, or reported infection by transfusion of this or any other coronavirus.