Intelligence officers sometimes have clandestine meetings, even in DC. Sec. Gates had at least one such meeting with then President-Elect Obama, before it was decided he would remain SecDef in the Obama administration.
As for FBI and CIA catching spies, I point to Robert Hanssen, who was brought down by his own arrogance. As ironic as it sounds, he was brought down by a system he devised while working counter-intelligence at the FBI. Hanssen came up with a system for ranking foreign operatives by what they have access to, not by their rank. When they were searching for the KGB mole, they checked who could have accessed the compromised reports. In some cases it was actually a small number of people had the access, including Hanssen.