"Do I trust him?" Bush said of Putin, standing alongside the Russian leader on the craggy Atlantic shoreline. "Yes, I trust him."
"There are times when we've agreed on issues and there are times when we haven't agreed on issues."
"The relationship of our two countries would be raised to an entirely new level," Putin said.
The Russian president renewed his opposition to the US missile defence plan.
He said he was not convinced by US insistence that the shield, with components based in Poland and the Czech Republic, was meant to defend against "rogue states" like Iran.
Putin issued a new proposal, expanding on his offer last month for joint use of Russian radar in Azerbaijan as an alternative.
"We support the idea of consultations on missile defence and believe that the number of participants should be expanded to include the European states," Putin said.
"This should be done within Russia-Nato council," he added. He suggested information-sharing centres in Moscow and possibly Brussels as a way to strengthen the Washington-Moscow national security relationship.
Bush called the Russian leader's latest missile defence idea "very sincere" and "very innovative".
Other key areas of disagreement between the two countries include the future of Kosovo and the state of democracy in Russia.
Bush, pushing for tougher sanctions against Iran, said Putin shared US concerns about Iran's nuclear programme.
"I've been counting on the Russians' support to send a strong message to the Iranians," Bush said.
Bush would not say whether or not the US and Russia have agreed on tougher sanctions against Iran, saying only that the two countries stand together.
"When Russia and the United States speak along the same lines, it tends to have an effect and therefore I appreciate the Russians' attitude in the United Nations," Bush said. "We're close on recognising that we got to work together to send a common message."
Putin predicted that "we will continue to be successful" as they work through the UN Security Council.
Both sides had advised against expecting breakthroughs at the end of the two-day informal summit.
Instead, they portrayed the meeting as a chance for Bush and Putin to revive their relationship.