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."

When Putin meets Biden on Wednesday in Switzerland, experts in Moscow say for all their differences, the two leaders want something similar from their first summit: to cool things down.

The U.S. and Russia's relations are the worst they have been since the Cold War and since 2016 in particular seem locked in almost permanent crises.

Biden has said he wants a more stable and predictable relationship with Russia, one that would allow it to focus on other foreign policy priorities that are more important to it, like taking a harder line with China. The Kremlin for its part has faced a continuous and intensifying barrage of sanctions-- the latest in April-- and with its crackdown on opposition at home and aggressive actions abroad is increasingly becoming a pariah with western countries.

Since coming to office, Russia has appeared to want to get Biden's attention. The president offered Putin the summit after Russia massed thousands of troops on Ukraine's border in April.

But now, having got Biden to the table, analysts said Putin has a clear proposal to deliver in Geneva: stay out of Russian domestic politics and Russia might act less troublesome abroad.

"The Kremlin wants to transition to a respectful adversarial relationship from a disrespectful one we have today," said Vladimir Frolov, a former diplomat at Russia's embassy in Washington and now a commentator on foreign affairs.

"That is, it wants to be treated the same way the Soviet Politburo was treated by the US in 1970-80s," Frolov told ABC News. "Meaning no name-calling" — such as Biden calling Putin a "killer" — "no personal sanctions on the leadership, no democracy lectures, regular personal summit meetings; respectful tone of discussions, no tangible support for Russian opposition."

It will not be an invitation for détente but instead to return to the later years of the Cold War when Putin was a KGB agent and the Soviet Union and the U.S. saw each other as enemies but tried to maintain a predictable relationship. And, crucially, where Russia was treated as an equal.

"For this, the Kremlin is prepared to promise to behave more responsibly," Frolov said.

"This seems to be in line with what the White House sees as a desirable deliverable," he continued. "So unless one of the leaders stormed out of the meeting shouting expletives, the summit would be a major success."

-ABC News' Patrick Reevell