In 1960, the U.S. had limited human intelligence on the ground inside the Soviet Union, and desperately wanted more information on its military capabilities.
"So we had spies in aircraft that could take pictures of the Soviet Union," said Costa.
One of those pilots was Francis Gary Powers, who was working for the CIA and was flying a U-2 spy plane 70,000 feet above the Soviet Union when he was shot down.
The Americans didn't think the Soviets could take down a plane at that altitude, and they didn't expect a pilot would survive such an emergency.
They were wrong on both counts, and the result was high drama at the height of the Cold War.
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A proposal to free two Americans
Today, the U.S. and Russia are trying to work out a prisoner swap that involves American basketball star Brittney Griner and former Marine Paul Whelan, apparently in exchange for Russian Viktor Bout, a notorious weapons dealer who's serving a 25-year sentence in Illinois.
While Washington and Moscow have done deals for decades, they usually involved trading spies for spies. The new twist is that an increasing number of private citizens are becoming entangled in foreign legal systems.
"During the Cold War, it was very much understood by both sides. This was a bit of a gentlemen's game," said Costa. Before running the Spy Museum, he worked at the White House, where he dealt with cases of Americans detained or held hostage abroad.
"What we're seeing play out now is really more hostage diplomacy," said Costa. He describes Griner as a "pawn in an international game. She has pled guilty (to a drug charge) and yet she is being held for political reasons."
The old rules
That doesn't mean it was easy to broker a deal in the Cold War, but the dynamics were different.
"When my father was shot down, there were all sorts of misinformation, fake news being published as to how he was captured," said Francis Gary Powers Jr., the son of the pilot.
He's referring to the U.S. government and media, which genuinely didn't know how Powers' plane was taken down initially, or how the case should be portrayed publicly.
"They thought sabotage. They thought pilot error or they thought flame out. They thought UFO encounter," Powers said with a chuckle.
Three years before Powers was captured, the U.S. had convicted a Soviet spy, Rudolph Abel. He had posed as a photographer in Brooklyn, but was secretly passing along coded messages stuffed inside hollow coins.
The U.S. and Soviet governments each feared their own spy would spill secrets while being interrogated by the adversary.
"Our governments wanted to get them back, to be debriefed, to find out what happened. How did you get caught? Do the Soviets have the missile technology to shoot down the U-2?" said Powers, founder of The Cold War Museum outside Washington.
Meanwhile, the Soviets "wanted to debrief Rudolf Abel to find out how he got caught so that they could improve their intelligence systems in the future."
Director Steven Spielberg turned the story into a movie in 2015, called "Bridge of Spies."
Tom Hanks plays the American lawyer at the center of the negotiations. He makes the case that the convicted Soviet spy shouldn't be put to death, because if an American is captured, "we might want to have someone to trade."
That American turned out to be Francis Gary Powers. Still, Powers would be held for nearly two years.
Already difficult negotiations were further complicated by the U.S. insistence that a second American detainee be released. He was Frederic Pryor, a graduate student held in communist East Germany on suspicion of spying.
A deal was eventually struck, freeing the two Americans and the Soviet spy.
Powers died in 1977 in a helicopter crash in California. In 2017, Powers Jr. traveled to central Russia to visit the site where his father touched down by parachute. There he met a number of residents who still remembered the day an American pilot fell from the sky more than a half-century earlier.
A series of swaps
In those intervening years, multiple U.S.-Russia prisoner exchanges have been worked out, most involving actual or suspected spies.
The biggest such case was in 2010, when the U.S. exchanged 10 Russian spies caught in the U.S. for the freedom of four Russians who had been arrested in their homeland, accused of spying for the West.
Today the Biden administration describes the two jailed Americans as "wrongfully detained." The U.S. has stated publicly that it wants to make a deal, but the Russian leader Vladimir Putin seems certain to drive a tough bargain.
"He's not easy to deal with," said Oleg Kalugin, a former Russian spy who was Putin's boss in the 1980s, when Putin was a young intelligence officer.
"Putin is sly, smart, and he manipulates people and circumstances if he can," said Kalugin, who's 88. After decades in the Soviet intelligence agency, the KGB, he became a critic of the Soviet Union in its final days and eventually moved to the U.S.
He says he wasn't much impressed by Putin when he was his boss. Now, they really don't care for each other.
"He called me a traitor. I called him a war criminal," said Kalugin.
U.S.-Russia relations continue to spiral downward over the war in Ukraine and a host of other issues.
But there's still a broad consensus on the need to keep lines of communication open to deal with issues like prisoner swaps or potential military confrontations.
"Throughout the Cold War, there was a communication mechanism," said Costa. "There can be unintended consequences of putting U.S. military forces close to Russian military forces. So it's absolutely crucial that those channels of communication remain open."