Iraq's Military Elite Among World's Best

They are Iraq's most secret and elite warriors, trying to save their country, fighting side-by-side with their U.S. mentors.

They are Iraq's version of the U.S. Delta Force, going night after night with their U.S. mentors into the most dangerous areas, hunting down the most wanted as they did when CBS News chief foreign correspondent Lara Logan joined them on this mission targeting a suspected Shiite sniper cell in Baghdad.

Logan's rare glimpse into their shadowy world revealed something startling: After four years of hard training, hard fighting and thousands of missions, U.S. Green Berets and Navy Seals have built an Iraqi counter-terrorism force that rivals any in the world.

"The way they surgically take down these targets, the way they treat everyone with respect and dignity, they are phenomenal and I would put them up against anybody," says a Green Beret commander.

"Even Americans?" asks Logan. "Even Americans," he replies.

This is Iraq's other war, against Shiites accused of killing U.S. and Iraqi forces. Logan says what it's really about is money and position in the future Iraq, after the U.S. leaves. As the Shiites vie for power, anticipating a U.S. withdrawal, much has changed for these elite soldiers.

At first they were condemned for raids against Shiite targets, like a hugely controversial one in Sadr City more than a year ago that left more than 20 dead. They were labeled the "Dirty Squad" by Iraqi leaders, including Prime Minister Nouri al-Maliki, who repeatedly attacked them for going after his Shiite allies.

Now, as he clings to power, it's the prime minister himself who calls on these forces time and again, even sending them back into Sadr City on a hostage rescue mission to try find the deputy oil minister who was kidnapped late last month.

They didn't find him, and came under heavy fire as they left Sadr City; a sign they're still not welcome in the militia-controlled area.

Iraq's best soldiers are caught in the middle of constantly shifting alliances that make this war increasingly complicated for them - and for the United States.