NEW YORK — One is a party luminary who draws intense political speculation wherever she goes. The other is the vice president of the United States, harboring his own higher ambitions. Democrats Hillary Rodham Clinton and Joe Biden appeared together Wednesday in New York, an event putting on display the unusual dynamics between these two friends and sometime-rivals.

The likely subtext at the Clinton Global Initiative’s annual awards ceremony, where Biden and Clinton honored citizens who tackle problems around the globe: whether either of them could someday win the ultimate prize they each sought in the past — the White House.

Biden, before presenting an award to New York Mayor Michael Bloomberg, acknowledged their friendship. “We miss you,” Biden told Clinton from the podium. “I miss our Tuesday morning breakfasts.”

Both have a base of loyal supporters encouraging them to run again, and neither has ruled it out. Many Democrats suggest the former first lady, who does not hold political office, would have a heavy advantage over Biden, who ranks second only to the president among the nation’s elected Democrats, should both get in the race. Until they each decide whether to run, such joint appearances will make for interesting political theater.

“It’s like a Rorschach inkblot. People will read into it what they want to see,” said Maria Cardona, who advised Clinton’s 2008 presidential campaign. “Most people would think, ‘How uncomfortable must it be to be in the same room with them!’ Probably not for them. It’s a comfort, because they’ve already been in those roles before.”

Clinton and Biden share a long history that has evolved over decades. They have been Senate colleagues, 2008 primary opponents, and principal players on foreign policy in the Obama administration. And their alliance dates back to Bill Clinton’s presidency, when Biden used his perch on the Senate Judiciary Committee to help pass legislation targeting crime, gun violence and domestic abuse.

During the dinner, the Clintons sat next to the vice president and his wife, Jill Biden. Those who have worked with Clinton and Biden say they’re genuine friends who enjoy each other’s company. While no longer working together in a formal capacity, their status as two of the highest-profile members of their party ensures their paths cross frequently.

In July, the two dined on eggs, bacon and fresh fruit at the vice president’s residence at the Naval Observatory, which is within walking distance of the Clintons’ Washington home. The private breakfast came a day after Clinton had lunch at the White House with President Barack Obama, stoking speculation the president might be signaling his support for his former secretary of state in the next presidential race.

A month earlier, Biden and Clinton spoke in New York at the funeral of Sen. Frank Lautenberg, D-N.J. And in April, they shared the stage at another awards ceremony at the Kennedy Center, a few blocks from the White House, where they praised each other like old chums.

“Vice President Biden and I have worked together on so many issues,” Clinton said.

“There’s no woman like Hillary Clinton,” Biden chimed in.

A private citizen after resigning as the top U.S. diplomat, Clinton has stepped back onto the national stage recently, appearing refreshed and energized as she speaks before groups that include key Democratic constituencies, including college students, black women, and the gay and lesbian community.

Biden, meanwhile, has become the president’s most visible surrogate, heading up efforts to curb gun violence, promoting trade and infrastructure, and traveling the globe to meet with world leaders. He also has kept up his profile in Democratic circles. An aggressive schedule of fundraising and campaigning has taken him repeatedly to important presidential primary states like Iowa and South Carolina.

Although well-liked in New Hampshire — another key primary state — Biden would have a tough time competing with the draw of potentially electing the first female president, said Sylvia Larsen, the Democratic leader in the New Hampshire state senate who advised Clinton’s 2008 campaign there.

“I don’t think the cadre of folks who would walk the plank for him are as strong as they are for Hillary,” she said.

That Biden may have to wait for Clinton to announce her plans before assessing the viability of his own candidacy is not lost on the vice president’s team. But John Marttila, who has advised Biden throughout his political career, said both Clinton and Biden are in the enviable position of being able to lay the groundwork early without having to commit now.

“It’s just due diligence for any prospective future campaign,” Marttila said. “For Biden, this is just like fun. I know he enjoys campaigning very much.”

Obama inevitably finds himself in the middle. The president has taken pains not to look ahead to 2016, wary of hastening his own lame-duck status. A joint interview with Clinton on CBS’ “60 Minutes” at the end of her tenure as secretary of state stoked speculation that he was tipping his hand about his preferred successor — a move that irked some in Biden’s circle.

“We are tremendously lucky to have an incredible former secretary of state, who couldn’t have served me better, and an incredible vice president, who couldn’t be serving me better,” Obama said this month in another interview. “I just got re-elected last year. My focus is on the American people right now. I’ll let you guys worry about the politics.”