Regional trains: tickets must be scanned, either from a piece of paper or an electronic device.
High speed and Intercity trains (those with reserved seats): the conductor only needs to know the PNR code you received via mail. You can show it using a smartphone, write it on the back of your hand or learn it by hearth: the conductor couldn't care less.
You need to be cautious of purchasing without registering for 4 reasons:
- you can't make any change on-line
- you won't receive a fiscal invoice. Those who are on a business trip often need one.
- if the train is severely delayed and it's trenitalia's fault, you can't ask on-line for the refund set by the EU law
- if you delete the e-mail, loose both the print out and your memory, the TSA cuts your hand off and you also forget the smartphone/tablet/PC at home... you won't have a way to know your PNR code and you'll have to purchase a new ticket on the spot.
I think you can register on trenitalia.com using a mail address in Italy to receive Trenitalia's fidelity card. If not it would be discriminatory towards other EU citizens: by registering you get access to more discounts.