The Italian government is tightening up on the borders. Within a few days anyone entering from some countries will have to have a recent certificate that they are Covid free or undergo tests at the border crossings. They are trlaling tests at Malpensa airport that give results in 30 minutes. I suspect that within a short period they will go back to banning cross border movement for non-essential reasons.
At this moment I would suggest that anyone thinking of travelling to Italy should keep a close eye on the situation. Personally I have not and will not cross any borders until I get a vaccine or it is otherwise all over.