We are not big revelers, but I can tell you the standard NYE festivities.
There is a huge gathering in Piazza del Popolo that I am told should be avoided as people tend to get out-of-control. (An FBI agent told me this and he is no sissy so I would heed his advice.)
In Via Fori Imperiali there is always a throng between Piazza Venezia and the Colosseo. There is a concert (free) there. There are fireworks over the Colosseo. (Also fireworks at Piazza del Popolo but as mentioned, might be dangerous.)
Some years there is a concert at Circo Massimo.
Most Romans simply go out for a nice -- and very expensive -- fixed price dinner, often starting at 10:00PM or so, and always staying past midnight. Reservations are a must! The Valadier restaurant overlooks Piazza del Popolo and I understand you can see the fireworks from there, but expect to pay €€€€.
Keep an eye on the site www.wantedinrome.com. They usually have info leading up to whatever the final events are that the city plans.