Only visitors from certain countries get tested at FCO. I think they're Croatia, Romana, Greece and Spain, but I haven't checked lately.
I arrived from the U.S. a few weeks ago and had to fill out a form at FCO, after having done so in New York, stating where I'll quarantine and promising to contact the local health authorities. Usually the passport check for Italian citizens is cursory and, more recently, we just scan our passports. But this time the agent did a mini-interview, made sure I understood the quarantine requirement. He also stressed that I needed private transportation to get where I was quarantining.
In general, it was pleasant and quick. Italian airport people tend to be more soft-spoken and polite than the overburdened and underpaid TSA workers in the U.S. Plus, I got my bag quickly because the plane was maybe 1/4 full.