This is directly from Princess, so don't shoot the messenger, please. FAQ: Cruise Answer Place : What To Bring : Princess Cruises
Click on Clothing Recommendations, and under Formal you'll find, "tuxedo, dark suit or dinner jacket and slacks for men."
So, it sounds like Princess does, at least ask for a jacket. Then again, it's listed as a suggestion, not a requirement -- and BritChris' experience echoes that.
So, I'd say to bring the jacket, wear it into the dining room, and either unbutton it when you sit down or follow the lead of the others. No reason why the same jacket can't be worn on both formal nights. I've actually worn the same gown on both formal nights.