You need to clear security for each and every flight. But, that is the wrong question. The question is how far in advance do you need to check in for your flight from Rome? Your airline should tell you this, but 1 hour in advance is typical.
I am assuming you are doing this as one booking with one ticket. In this case you would not have been sold the ticket unless there was enough time for the transfer. 1 hour is just enough time at Amsterdam; but, if your first flight is delayed, it is the airlines responsibility to put you on the next flight.
The next question is emigration and customs. If you have a through ticket your luggage will be labelled through from Rome to your final destination. There is never customs on exit, so the next time you will see your luggage is in the US, and you go through customs there.
Rome to Amsterdam is internal to the Schengen Area, no passport checks (but the airline will of course need to see your passport for ID checks). You exit the Schengen Area at Amsterdam, your passport will be checked there and if necessary stamped. US immigration will take place when you land there.