Bush also conferred with Putin's hand-picked successor, Dmitry Medvedev, but did not claim gaining any insight into his soul, as he had with Putin upon their first encounter. He pronounced Putin's protege "a straightforward fellow" and said he was eager to work with him.
Putin was asked whether he -- or Medvedev, the president-elect -- would be in charge of Russia's foreign policy after May 7, when Putin steps down as president and is expected to be named prime minister.
Putin said Medvedev would be in charge, and would represent Russia at the Group of Eight meeting of industrial democracies in July in Tokyo. "Mr. Medvedev has been one of the co-authors of Russia's foreign policy," Putin said. "He's completely on top of things."
National Security Adviser Stephen Hadley, when asked later whether he thought Putin actually was going to cede authority on Russian foreign policy to Medvedev, said, "My guess is that these two men who have worked very closely together for now almost two decades will have a very collaborative relationship. That seems to be a good thing, not a bad thing."
Hadley, who spoke with reporters aboard Air Force One on the way home to Washington, also said he didn't see any prospect of a breakthrough on missile defense before Bush leaves office next January. "They can leave that to their prospective successors," he said.
At their 28th and presumably final meeting as heads of state, Bush and Putin sought to emphasize their good personal relations, praising each other extensively. But they also both acknowledged remaining strong disagreements, principally missile defense and NATO's eastward expansion.
Russia remains adamantly opposed to the expansion of the alliance into its backyard, an enlargement that Bush has actively championed over Putin's vocal objections.
The Sochi meeting came just days after NATO leaders agreed at a summit in Romania to invite Albania and Croatia to join the alliance. However, the alliance rebuffed U.S. attempts to begin the process of inviting Ukraine and Georgia, both former Soviet republics, to join, although their eventual admission seems likely.
Putin called the U.S. missile plan -- which envisions basing tracking radar sites in the Czech Republic and interceptors in Poland -- the hardest of US-Russian differences to reconcile. "This is not about language. This is not about diplomatic phrasing or wording. This is about the substance of the issue," he said.
Bush reiterated his insistence that the plan -- designed to intercept and destroy approaching ballistic missiles at high altitudes -- should not be viewed as a threat to Russia. In a clear reference to Iran, he said the system would help protect Europe from "regimes that could try to hold us hostage."
"I view this as defensive, not offense," Bush said. "And, obviously, we've got a lot of work to convince the experts this defense system is not aimed at Russia."
He blamed opposition to the plan to lingering Cold War fears.
The two leaders did issue a joint statement on missile defense as part of a "strategic framework" to guide future relations between Washington and Moscow.
The statement outlined timeworn U.S. and Russian positions but also held out the prospect for future cooperation, perhaps on a joint system. That, said Putin, represents "certain progress."
"If we manage to achieve this kind of level of cooperation on a global missile defense system, this will be the best kind of result for all our preceding efforts," he said.
Bush bristled at a journalist's question that suggested the two leaders were merely "kicking the can down the road" on the vexing issue.
"You can cynically say that it is kicking the can down the road. I don't appreciate that, because this is an important part of my belief that it is necessary to protect ourselves," Bush said.
The two sides also agreed to "develop a legally-binding arrangement following expiration" in December 2009 of the strategic arms limitation treaty (START). Their joint declaration noted the "substantial reductions already carried out" under that pact, which they said was an important step in reducing the number of deployed nuclear warheads.
Bush was reminded of his June 2001 comment after his first meeting with the Russian leaders that he had looked into Putin's eyes, "was able to get a sense of his soul" and found him to be trustworthy. The remark startled even some of Bush's own aides at the time.
"I did find him to be trustworthy, and he was trustworthy," Bush said Sunday. "He looks you in the eye and tells you what's on his mind. He's been very truthful. And to me, that's the only way you can find common ground."
And did he feel the same way in his first meeting on Sunday with the next Russian president?
"I just met the man for 20 minutes," Bush said. Still, Bush said Medvedev "seemed like a very straightforward fellow. My first impressions are very favorable."
"You can write down: I was impressed and looking forward to working with him," he told reporters.
Bush met separately with Medvedev before his news conference with Putin and received a pledge from the incoming president to work to strengthen relations between the two countries.
Over the last eight years, Bush and Putin "did a lot to advance U.S.-Russian relations" and that relationship was "a key factor in international security," Medvedev said. "I would like to do my part to keep up that work," he added.
Bush told Medvedev, "I'm looking forward to getting to know you so we'll be able to work through common problems and find common opportunities."
A bond of sorts formed between Bush and Putin when Putin stood with the United States after the Sept. 11 terrorist attacks. But the era of cooperation quickly began to unravel.