Their comments, in fielding questions from university students and faculty at St. Petersburg University, also revealed some economic tensions.
But disagreements were few in the good-natured and often playful exchanges between the two leaders in Putin's hometown. Mr. Bush called Putin "my friend" and referred to him as "Vladimir" several times. Putin called him "George."
During the bantering, each suggested the audience send the hard questions to the other.
The event, similar to one the two held in Texas last year, was broadcast live on Russian television. Putin graduated from the school in 1975 when it was known as Leningrad State.
Both presidents saluted the demise of the Cold War. "That era is long gone, as far as I'm concerned," Mr. Bush said. "Today America and Russians are friends."
"We've got a new war to fight together," Mr. Bush said when asked about the role of force in the future relations between the two countries.
He said that Russia and the U.S. would combine forces "to fight against bloodthirsty killers."
Putin blamed some Cold War-era U.S. trade restrictions for making it harder for Russia to export high-tech goods. And Mr. Bush said that, while he supports Moscow's entry into the World Trade Organization, he opposes bending the group's stiff standards to make it happen.
"I think the accession to the WTO ought to be based on the rules every other nation has had to (follow) ... nothing harsher, nothing less harsh," Mr. Bush said.
Commerce Secretary Don Evans will decide within weeks whether to declare Russia a "market economy," a designation important for its entry into the Geneva-based organization that sets and polices world trade rules.
It would represent approval from the United States, the WTO's biggest member, of Russia's efforts to move from a government-controlled economy to the free market. Membership itself will depend on Russian negotiations with individual WTO members over how it will lower trade barriers.
The decision comes at a time of trade tensions between the two countries over U.S. tariffs on Russian steel and Russian restrictions on U.S. poultry products.
A day after signing one of the most sweeping arms-reduction pacts in history, the two leaders heralded an era of new good political and economic will.
"A strong, prosperous and peaceful Russia is good for America," Mr. Bush said.
Putin praised the treaty and a second pact that outlines a new strategic relationship between the United States and Russia.
At the same time, his remarks recognized that there was some opposition within Russia to the pacts, particularly from some elements of the military and other hard-liners.
In Mr. Bush's recorded weekly radio address Saturday, he praised the summit accomplishments and Russia's warming relations with the United States and Europe.
"President Putin and I are putting the old rivalries of our nation firmly behind us, with a new treaty that reduces our nuclear arsenals to their lowest levels in decades," he said. "After centuries of isolation and suspicion, Russia is finding its place in the family of Europe. And that is truly historic."
At a news conference, Secretary of State Colin Powell said the two countries would keep trying to resolve differences over Russia's nuclear assistance to Iran. The United States is concerned the assistance could help Iran develop nuclear weapons; Russia insists the technology is being used for nonmilitary purposes.
"I hope that we will be able to solve this going forward," Powell said.
Powell also expressed concerns about Russia's short- and medium-range nuclear weapons, which were not covered by Friday's agreement. "We still have some, they have many more," he said.
But he focused on the positive, saying the summit had made the world safer by cutting nuclear arsenals and strengthening personal ties between U.S. and Russian leaders.
One sign of the warming relationship was that most students' questions were on economic issues.
Asked why Russia's biggest exports were basic products like oil and wood rather than high-tech products, Putin got in a gentle dig at U.S. restrictions that he said discriminated against Russian products.
"We need nondiscriminatory access to world markets and U.S. markets," he said. "We don't want preferences ... we don't want special favors."
Mr. Bush reiterated his support for repealing the Jackson-Vanik trade law of the 1970s that ties Russia's trade status to its progress on Jewish emigration. The repeal is bogged down in Congress.
Mr. Bush praised Russia's flat tax as fairer than taxes in some Western countries. But he said Russia's export tax worked against its own interests.
He also said it was a good sign that the percentage of private ownership of business in Russia had climbed to 70 percent.
One questioner asked Mr. Bush what specific steps were needed for Russia to join the WTO.
"Starting with having a president who thinks you should be in the WTO. And I think you ought to be," Mr. Bush said. "I vote `aye,' assuming that the Russian government continues to reform their economy ... and make a market-based economy work."
WTO membership would make Russia a more predictable place for foreign investment. But difficult economic changes are needed, and some U.S. trading partners have voiced concern that Russia not be given exemptions from those rules.
"George said it very well. The president of Russia has to want to be a member of the WTO. And he said that he's for it. If that's sufficient, I'm in," Putin said, drawing laughter.
Anti-globalization activists were among a few hundred protesters who followed Mr. Bush on Saturday. Leaders of the protests, which included Communists and nationalists, were driven away by plainclothes security personnel. The protests were small compared with those attended by 20,000 demonstrators when Mr. Bush visited Berlin on Thursday.
He arrives in France Sunday, and thirty French organizations have joined to plan fresh demonstrations, including two high-profile protests on issues ranging from a possible U.S. attack on Iraq to Washington's polices on the Middle East, trade and the environment.
The university forum was the highlight of a day in which the presidents took in Russian sights and culture.
They laid a wreath of yellow and red roses at the Monument to the Motherland in Piskarevskoye Cemetery, which contains the mass graves of some 600,000 victims of the 900-day Nazi siege of Leningrad during World War II.
Afterward, they toured the Hermitage museum, the largest art museum in Russia. Walking arm-in-arm with their wives, Mr. Bush and Putin climbed the marble staircase just inside the entrance to see masterpieces by Da Vinci, Rembrandt and others.
In the evening, the Bushes and the Putins attended a performance of "The Nutcracker" ballet and then went on an evening boat cruise on the Neva River.
Because the city is so far north, there was still some light in the sky as they ended their cruise near midnight - to a celebratory display of fireworks.
Mr. Bush rarely is up so late, but since he and Putin were in such upbeat moods after what they viewed as a successful summit, he amended his early-to-bed routine, said White House spokesman Ari Fleischer.
"Since it's still light, we don't count it as midnight," Fleischer said.