Private transfer or the train?
From Fiumicino the Leonardo Express reaches Roma Termini every 15 minutes and costs €14 a person, running from 05:38 to 23:27 out of the airport.
The ticket covers the part between two stations. It leaves out the walk to the platform with the suitcases and the last stretch from Termini to your address, which is where a family with luggage gives back the time it saved.
A private transfer is one fixed price for the whole car, from the arrivals hall to the address you give. Four people pay once, not four times.
