Alternately joking and poking at each other, the two also showed differences at a news conference on the explosion of violence in the Mideast.
Mr. Bushfor its punishing attacks in Lebanon and said it was up to the militant group Hezbollah to lay down its arms. Putin was critical of Israel's use of force and said the violence "should stop as soon as possible."
The two leaders met for two hours before the opening of the annual summit of eight major world powers, which was expected to focus on nuclear problems with Iran and North Korea and the escalating fighting between Israel and the Islamic guerrilla group Hezbollah.
There was a quick handshake but little warmth between Mr. Bush and Putin during a photo opportunity opening their talks. For the second day, Mr. Bush spent part of it mountain biking.
Despite the sparring, there was none of the tension and anger that crackled in Bratislava, Slovakia, 17 months ago when Mr. Bush challenged Putin over Russia's crackdown on dissent and retreat from democracy and the Russian president slapped back. After that jarring meeting, Mr. Bush concluded that lecturing Putin in public was unproductive. Still, Mr. Bush said he offered Putin some suggestions.
"I talked about my desire to promote institutional change in parts of the world like Iraq where there's a free press and free religion," Mr. Bush said at the news conference, "and I told him that a lot of people in our country would hope that Russia would do the same thing."
Putin, in a barbed reply, said: "We certainly would not want to have the same kind of democracy as they have in Iraq, I will tell you quite honestly." Mr. Bush's face reddened as he tried to laugh off the remark. "Just wait," Mr. Bush replied about Iraq.
Putin also said Russia would not take part "in any crusades, in any holy alliances," a remark seemingly intended to win points with Arab allies. Mr. Bush's national security adviser, Stephen Hadley, said he was perplexed by the comment.
Hosting the Group of Eight summit for the first time, Putin dearly wanted to win approval for Russia's admission to the World Trade Organization, the 149-nation group that sets the rules for world trade. The United States is the only country that has not signed off on Russia's membership in the WTO, and Bush dashed Putin's hopes for getting in now.
"We're tough negotiators," Mr. Bush said, adding that any agreement would have to be acceptable to the U.S. Congress.
U.S. Trade Representative Susan Schwab said "significant progress" had been made in narrowing differences over the protection of U.S. copyrights and patents and boosting the sale of American manufactured goods. She said negotiators were unable to resolve a dispute over Russian barriers to the sale of American beef and pork. She said the hope was that the agreement could be completed "in the next couple of months."
Mr. Bush said he and Putin agreed on the need for the U.N. Security Council to take action against Iran for refusing to halt uranium enrichment and against North Korea for testing a long-range missile and refusing to rejoin six-nation disarmament talks.
"There is common agreement that we need to get something done at the U.N." on Iran, Mr. Bush said. He said there was common ground on North Korea, as well, and "now we're working on language." Russia and China have been reluctant to impose penalties on North Korea or Iran. Mr. Bush declined to say whether he asked Putin to back U.N. sanctions against Iran.
Lacking a centerpiece agreement on trade, the United States and Russia announced several lesser agreements:
Meanwhile, protesters challenged Russia's determination to keep a lid on demonstrations during the summit.
About 150 protesters faced off with police today in St. Petersburg when police refused to let them march. Russian authorities have prohibited marches and limited the main protest activity to a stadium in a hard-to-reach part of the city.
Another 250 people attended a rally organized by the Communist Party in the city center. They had permission to demonstrate.