Thank you for your question, I hope my answer will be useful to you!

Perhaps this is the great paradox about espionage and the business of what they call counter-intelligence. While spying from embassies is strictly forbidden under international laws that regulate the system of embassies and diplomatic representation, they're ideal forward stations inside other countries from where countries can launch clandestine operations under a relatively safe roof with guaranteed immunity.

International laws ensure that resident countries have no authority to enter the premises of diplomatic representations without their permission, search diplomatic bags or mail, arrest or detain diplomats and so on. So, of course, during the Cold War, every year the Soviet embassy would invite their "friends" of the local Communist party to a reception at the embassy, take them to a backroom, and hand them over a big bag of money - a diplomatic bag of money - to be spent on furthering the cause of world revolution. How rude of them to abuse the system like that!

Luckily, everyone knows and expects this to happen. So when our Fellow Travellers, the members of the Soviet-British friendship society, come walking out of the embassy reception - perhaps a tad tipsy, you can reliably trust on it that the black hats of Her Majesty's Security Service MI5 are sitting in a car, playing a game of spotting birds and ticking off every single name that attended that party. Those secret dealings inside the embassy never remain secret for very long - thats the idea at least.

It has happened in the past that a US Navy officer walked into the Soviet embassy and carried on spying with his entire family for the Soviets for more than 15 years before the FBI finally noticed something was wrong.

On top of that, embassies during the Cold War were also used as listening stations. Since the building is protected, nothing can really prevent that its upper floor is stuffed with interception equipment and satellites to quite literally eavesdrop on the building next door, which tends to be a government building of the resident country. The KGB used this to eavesdrop on the Pentagon, the CIA to listen in on the Kremlin. Perhaps this is why the CIA is not located in Washington with their headquarters, and the KGB's foreign departments in similar fashion reside in Yasenovo out of all places.

For the CIA, this was almost the only way to get access to the hermetically closed off world behind the Iron Curtain. They tolerated the risk of Soviet espionage against the US, in exchange for the ability to operate from the capitals of Moscow, Prague, Budapest, Warsaw etc. Besides, it was actually a beneficial trade off since the absence of embassies is no guarantee against spying.

On the contrary, when the Soviets lacked diplomatic representations in the 1920's, just as the East-Germans in the 1950's and 1960's, they resorted to the highest arts of conspiracy by what they called "illegals". Arguably, the most succesful spy penetrations of the Cold War were achieved through these Illegals - people trained for years to operate independently from any kind of embassy, to emigrate abroad, emigrate again, and build up a completely new life, and finally arrive in the target country. With their limited resources, Western security services could not be expected to place surveillance on every immigrant, tourist or foreign businessman entering the country. And so entire networks could be established by seemingly unsuspicious citizens from their living room, whose attic contained a radio set and served as KGB station for decades. These networks penetrated the highest government circles of the British, West-German, American and Canadian governments, and one of their achievements was stealing the atomic bomb designs.

From an FBI or MI5 perspective, its safer to have a Soviet embassy - so at least they know where to look. Without an embassy, their espionage activities will go completely underground and a good amount of luck was required to catch them. Or a Soviet defector. As a former CIA official said, it takes a spy to catch a spy.

For Moscow, it was also a quid pro quo. Having Western embassies meant having Soviet embassies - forward posts - in the West. They formed a badly needed window to the West. In the spy business, the Soviets really needed their embassies for SIGINT collection against Western countries. The US and UK together employed a network across the globe of eavesdropping stations and satellites, enabling to watch and eavesdrop on the Soviets from distance. The Soviets severely lacked behind in that respect, and needed to rely on naval vessels trawling past the US and British coasts, Cuba, and its embassies in Western capitals.

The KGB too calculated that the cost of hosting a US embassy in Moscow could be worth it. First of all, they made full use of their home advantage by manipulating the construction of the US embassy building through adding hollow walls and planting listening devices inside the building structure. The KGB revealed to the US after the Cold War what the 'real' building design looked like. On top of that, with the KGB employing some 400,000 people, they had enough manpower to literally shadow every foreigner inside their territory.

An example is the case of Oleg Penkovsky. He was a GRU colonel who had approached the West during the hottest years of the Cold War, because he believed the world headed to war. He wouldnt be the first nor the last to betray his country by a missionary conviction that he was saving the world. Penkovsky was eventually caught however, by what the KGB claims was routine counter-intelligence. Penkovsky was supposed to meet his handler in Moscow, a woman from the British embassy. She was also the wife of the embassy's MI6 station chief. The KGB shadowed the woman to a flat, where they recorded the face of the man she briefly met. Then the KGB spent the next months looking for that man again, until, by coincidence, they spotted him again and followed him straight to the headquarters of the GRU.

In the months that followed, they kept surveillance on Penkovsky until they could catch him in the act, and sentenced him to death. This story is almost certainly not true, but it shows how in the "ideal" situation, counter-intelligence benefits from having and watching foreign embassies. The story is disputed from all angles since 1991, because of many elements, and was probably designed to protect the fact that Penkovsky was fingered, betrayed, by a mole inside MI6, the British embassy, or the CIA. The first lesson in counter-intelligence is not to become too proud of the victory and never to tell the full story of how a spy was caught.

Because monitoring of embassies was also a numbers game, the Soviets tried to escape surveillance by quite literally overwhelming Western surveillance with agents. Consulates were almost entirely made up of KGB and GRU officers, embassies consisted of hundreds of people - half of whom worked as intelligence, and trade representations were also thinly disguised GRU stations. This took on such ridiculous forms that by 1971, the British government seized the defection of a KGB London officer to expel some 105 intelligence officers from the Soviet London embassy. If one takes into account that all the Warsaw Pact countries did the same, and then there were also other countries such as China, Cuba, Israel and so forth playing hide and seek in Western capitals - then it becomes clear that manpower alone means that many intelligence officers could freely roam around without a baby-sitter.

So to cut this story short, in the spy world having embassies is a sort of trade-off. They enable them to operate more easily abroad, and they hope they can control the damage in their own country through rigorous surveillance. In this respect the Soviets had an advantage over the West because of sheer manpower, enabling them to monitor every Western diplomat in Moscow while outnumbering Western agencies in their embassies abroad. For the West however, it was still a way of keeping sight on what the Soviets were doing, and good training enabled them to shake off even the KGB once in a while - which was a preferred alternative to having nothing at all.

If you want some further reading recommendations, then I can recommend the following works:

Christopher Andrew, "The Secret World: A History of Intelligence" (2018). This recent publication gives a broad, long term view of how intelligence was conducted throughout history, written by one of the most knowledgeable scholars on the subject.

Peter Jackson (ed.), "Intelligence and Statecraft: the Uses and Limits of Intelligence in International Society" (2005), this contains a useful selection of essays and chapters on the history of intelligence, also on the development in the 19th century that led to the time-honoured tradition of using embassies as spy stations.

For a more focussed KGB perspective, see also: Christopher Andrew & Vasili Mitrokhin, "The Mitrokhin Archive: the Secret History of the KGB" (1999). Probably still one of the most complete histories of the KGB drawing on the files taken by KGB-archivist and defector Vasili Mitrokhin.