In a summer when the president is traveling across the country meeting with “ordinary Americans” under highly choreographed conditions, the Rome dinner shows another side of Mr. Obama. As one of an increasing number of late-night dinners in his second term, it offers a glimpse into a president who prefers intellectuals to politicians, and into the rarefied company Mr. Obama may keep after he leaves the White House.

Sometimes stretching into the small hours of the morning, the dinners reflect a restless president weary of the obligations of the White House and less concerned about the appearance of partying with the rich and celebrated. Freewheeling, with conversation touching on art, architecture and literature, the gatherings are a world away from the stilted meals Mr. Obama had last year with Senate Republican leaders at the Jefferson Hotel in Washington.

As Mr. Obama once said about the Senate Republican leader from Kentucky: “Some folks still don’t think I spend enough time with Congress. ‘Why don’t you get a drink with Mitch McConnell?’ they ask. Really? Why don’t you get a drink with Mitch McConnell?”