Do we go through some sort of customs screening on arrival at the Rome airport?
Nope. The exit I left, at least, you walk out either the " " side where there's a window and I suppose you meet with them and tell them what you are declaring, or you just walk out the "nothing to " side. It should be the declare side and nothing to declare side, I'm not sure why it isn't showing up! I've edited it but it still won't work, haha
But you will have Immigration AKA passport control -assuming you are flying from the US.
Exactly. They do check your passport, but it isn't like flying back in to the US when you have to fill out the information and go through a very different process.
Assuming that you are arriving on a flight from the U.S., you will go through both immigration (aka passport control) and customs. After passport control, you will walk through the "green" lane (assuming that you have nothing to declare) and by walking through that lane you have gone through customs.
Sounds simple. How long does it take?
IT depends on the airport and the length of the line when you get there at passport control. Generally it takes "not very long" to a "little while". ;-) If you have nothing to declare, you'll just walk through the "green" lane.
And if you luggage doesn't arrive, which happened to us when arriving in Rome (we had made a quick transfer from our US flight a Charles De Gaulle, we made it but our luggage did not), we got to walk right through. Luckily our luggage did arrive the next day. In reality, everyone seems to walk right thru the GREEN lane, even with their luggage if they have nothing to declare.