I don't think I'm exactly qualified to answer this, but I would recommend The Sword and the Shield which discusses KBG operations extremely in depth from WWII through the 80s.

I read it for fun, not for study, but I would say that the answer to your question was that people pretty much went ahead as usual. They were always convinced that their anti-bug techniques were good enough to detect the enemy's bugging devices. They were always convinced their intel departments were keeping them safe while the enemy's intel departments were being infiltrated.

I would, however, also add that on the Russian side of things there was rampant paranoia. Soviet Russia was constantly convinced that Western governments were conducting elaborate plots against them to the point where actual quality intel was turned away because the KGB was convinced it was purposely planted. Their five major spies in the post-WWII era were frozen from an intel standpoint even while they were handing over secrets about the atomic bomb, because the KGB was convinced the info was too good to be true and they had been turned. This paranoia led to a bit more concern about bugs from the Soviet end in the KGB and related intelligence departments, but the main thing I gathered from my read was that the notion of spies and intel did not exist in a vacuum or as a singular event. People were aware that bugs might exist, people were aware that spies might exist, which meant that there were people employed to detect bugs and to detect spies. Throughout the Cold War there were successful bugs and successful spies and unsuccessful ones. Overall, though the diplomats were more focused on diplomacy and the intelligence employees more focused on intelligence and anit-intelligence measures. There was some overlap, but also a lot of trusting that each side had their end worked out.

Again, sorry if this isn't informative enough, I've only read the subject for entertainment, not for study. I'm hoping to learn from being corrected.

One method was (and is) to use a specially built room that was brought into the embassy from the home foreign nation and then assembled by trusted people. Most embassies (not just American) have them. Many high-level political meeting places and organizations dealing in sensitive discussions that they need to keep secret and sequestered from potential spying/eavesdropping gear also build these special rooms specifically so they can discuss sensitive matters in secrecy, away from the threat of electronic or vibration-based eavesdropping. They are known as "quiet rooms" or a "SCIF" (pronounced "skiff") - a Sensitive Compartmented Information Facility.

These SCIFs can look just like a normal conference room or office, and are designed to be shielded to prevent electronic signals from entering or exiting. I think Hollywood and some fiction writers are responsible in large part for the image of a SCIF being some cool looking "floating" room suspended by cables above the floor or something, though that is not always the case or a requirement.

For example, in the film "The Sum of All Fears" (based on the Tom Clancy novel), there is a scene when the Director of the CIA (played by Morgan Freeman) takes Ben Affleck's character (Jack Ryan) inside a cube-like SCIF obviously built inside a larger room to brief some high-ranking politicians on the crisis in Russia. The reality though is that you could be inside a SCIF in a building somewhere and not even know it if it was built correctly.

A SCIF is just basically a well insulated and isolated room-within-a-room in many cases, with internal walls like a Faraday cage that prevent electronic signals from entering or exiting, and usually, if they have windows at all, they are multi-pane windows with small white noise generators located between the panes of glass to prevent SPL vibration-sensing through the glass and walls. White noise generators also can be placed in the floor and ceiling to prevent SPL eavesdropping between floors. Access to the SCIF is tightly controlled, and electronic jammers can also be employed, but the room is really usually not too visually impressive to the naked eye at all. Because they are built by, imported, and assembled in the embassy by the foreign nation who owns the embassy however, their security can be assured.