When we are getting close to the peak of the epidemic, the growth will no longer be exponential, but close to linear. A logistic curve looks exponential for about the first 3rd, close to linear for the second third, and then approaching the asymptote with a negative exponential growth for the last third.

Increasing social distancing will naturally lower the exponential growth rate, but unless the growth becomes linear, there is no reason to believe that herd immunity is anywhere close.

In the asian countries, we more or less know that they have been able to contain the virus, instead of waiting for herd immunity. That meanst that they must have found most of the infected cases (or at least quaranteened them). This gives a good indication of how many deaths are caused by a given number of infected. The numbers are not different enough from Italy to think that more than (at most) a couple of hundred thousand in Italy have reached immunity. If so, it means that we have about 2 more orders of magnitude of growth before herd immunity can be reached. That would be hundreds of thousands of dead, if not millions, just in Italy.