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Every federal agency and department has to deal with that tension. But Hayden said it's worse for intelligence agencies.
"In the intelligence community, particularly in the CIA and particularly when talking about covert action, you are operating on the outer edges of executive prerogative," Hayden said. "In our system of government, this relationship is always spring-loaded to be tense."
To what degree the two powers stand on equal footing on intelligence matters - or whether they even should - is a constant source of drama.
"Equality has not been achieved," said Suzanne E. Spaulding, a former House and Senate Intelligence staffer who now serves as a deputy undersecretary in the Department of Homeland Security. "Whether it's appropriate and what it should look like is the more interesting question."
Spaulding - speaking personally, not on behalf of the Obama administration - said balance, rather than equality, should be Congress' goal. Each side, she said, has different mechanisms of power: The administration has great power to carry out specific spy operations, while Congress has the power of the purse.
R. James Woolsey, CIA director from 1993 to 1995, said he knows the power of the purse: He said he had 205 appointments on Capitol Hill in 1993, more than the number of days Congress was in session. Mostly, he said, he was defending his agency's budget from looming cuts.
Historically, the most successful CIA directors are those who have possessed a long list of similar traits, among them good relationships with Congress, according to David S. Robarge, the CIA's chief historian.
One of the keys to a good relationship is proactive, thorough information sharing, Robarge said. Going back to Goss' problem - it doesn't always happen, and Robarge said nothing angers Intelligence Committee members more than hearing about a new intelligence operation first in the New York Times or Washington Post.
Congress also sometimes wants more information than administrations are willing to give, something Spaulding said leads to games of "20 questions" in intelligence briefings from staffers and lawmakers trying to wrench fundamental details out of intelligence officials.
And intelligence officials are sometimes reluctant to share information with Congress, because, Goss said, politicians are trained to talk, and talking equals "leaks" for spy officials. "We've not stopped the problem of how you stop leaks on the Hill," Goss said. "You never will."
Others, including Spaulding, maintain that Congress is not a significant source of national security leaks. And over time, Goss said, intelligence agencies have done a much better job of briefing the committees, which in turn leads to better oversight.
Better oversight doesn't necessarily always mean putting a halt to a president's plans, though, Spaulding said.
Congress has tried and failed to stop a variety of controversial post-Sept. 11 spy policies, such as harsh interrogations, or endorsed them, as in last week's House vote to extend the expiring surveillance authorities contained in a 2008 law that effectively authorized President George W. Bush's so-called warrantless wiretapping program.
"Some of those are instances where there was perhaps more oversight and give-and-take than was apparent to the public taking place in classified spaces that got us to a point where there was a stronger consensus in Congress to support some things in Congress that were very controversial at the outset," Spaulding said.