Biden and Putin's first -- and likely only -- photo op inside their summit was met with a chaotic scene.

According to a pool report, the chaos began outside the room with "10 minutes of a shoving match" between the U.S. and Russian security, press and delegations, with each side aiming to get inside. American reporters were first to go in but not all were let inside.

Already seated, Biden nodded while they shouted questions, including whether the two leaders can trust each other -- but the White House is already disputing that Biden was nodding yes to that.

"It was a chaotic scrum with reporters shouting over each other. @POTUS was very clearly not responding to any one question, but nodding in acknowledgment to the press generally. He said just two days ago in his presser: 'verify, then trust," White House communications director Kate Bedingfield said in a tweet.

Another reporter asked Putin if he feared imprisoned Russian opposition leader Aleksei Navalny and what he would do if Ukraine is allowed to join NATO. Putin didn't answer.

On the way out, another shoving match ensued.

"Lots of shoving and grabbing - it was extremely aggressive," one American pool reporter said. "The Russian security pulled on our clothes and shoved us as we tried to stay in the room. They eventually pushed us out the door."

Inside Villa La Grange, the high-stakes summit has officially kicked off.

Speaking to reporters briefly at the top of their first closed-door sit-down, Putin said he hoped for a "productive" meeting, and Biden said "it is always good to meet face-to-face."

While the scene was chaotic, both leaders looked comfortable. Biden, who was the first to extend his hand for a handshake earlier, sat with his legs crossed, hands in his lap and was seen smiling at several points. Putin also leaned back in his chair, as he often does.

Seated in a library before their respective country's flag and with a globe in between, the pair were joined by U.S. Secretary of State Antony Blinken and Russian Foreign Minister Sergey Lavrov for the first small session of the day.

Following brief remarks from Swiss President Guy Parmelin wishing them well, Biden and Putin shook hands in front of cameras, with both men grinning, before entering the summit site.

Biden and Putin's meeting is expected to last four to five hours total with multiple sessions.

First, they're taking part in a small session, joined by U.S. Secretary of State Antony Blinken and Russian Foreign Minister Sergey Lavrov, ahead of a larger working session.

Biden and Putin are also expected to hold dueling, solo press conferences following the summit.

Swiss President Guy Parmelin wished Biden and Putin "a fruitful meeting" on Wednesday, just minutes before the two leaders sit down for their much-awaited bilateral summit in Geneva.

"On behalf of the Swiss government, I would like to welcome you to Geneva, the city of peace," Parmelin said in prepared remarks in French while welcoming them to Villa La Grange, where the meeting is taking place.

"It is an honor and a pleasure for Switzerland to host you here for this summit and, in accordance with its tradition of good offices, promote dialog and mutual understanding," he added. "I wish you both a fruitful meeting in the interest of your two countries and the world."

Putin indicated on Friday that he'd be willing to talk about a potential "prisoner swap" between two U.S. Marine veterans, Trevor Reed and Paul Whelan, for Russians being held in the U.S.

Russian officials have indicated they would like to trade Reed and Whelan for two Russians held in the U.S.: Viktor Bout -- one of the world's most notorious arms dealers and dubbed "the Merchant of Death" — and also Konstantin Yaroshenko, a pilot currently serving a lengthy jail sentence for a drug smuggling conviction.

David Whelan told ABC News Live Prime anchor Linsey Davis on Tuesday that his brother, who is manufacturing clothes in a Russian labor camp, is a hostage and that it's difficult to know when that might come to an end.

"I'm always hopeful that he'll be released, but I don't have any idea what the timeline might be," David Whelan said. "The Biden administration has been very outspoken about Paul's case and we've appreciated that as a family. It's given us hope and it's given him hope."

"But he's still a hostage and there's no evidence yet that the Russians are willing to exchange in any exchange for him," David Whelan continued. "They said last week -- the Russian government did — that they would not consider Paul for exchange."

Whelan's family released an audio message from him on Monday recorded from the prison camp in central Russia where he is held. In it, he appealed to Biden to help free him.

"Please bring me home to my family and my dog Flora where I belong. Thank you, Mr. President, for your commitment to returning me home and bringing this deplorable hostage situation to an expedient conclusion," Whelan said in the recording that his family said was made on May 30.

-ABC News' Haley Yamada and Patrick Reevell