The two things to consider here are the Soviet dominance of human intelligence and a far superior Soviet counterintelligence units. So I will answer in two parts.

Soviet human intelligence vastly outpaced the CIA during the early to mid Cold War. The Soviet Union recruited young, ideologically motivated, foreign nationals (those in target states) with a high potential to reach important government positions. When many of these agents reached those positions it became clear that the CIA, MI5, and MI6 were not even under siege but instead had already become thoroughly penetrated. Author Wilderness of Mirrors, David C Martin says that "[the CIA] was utterly ignorant of Soviet espionage operations." Agents in the employ of the Soviet Union often installed themselves in the organization's core structure before rigorous internal counterintelligence programs were instituted. This means that the infiltrators were trusted members of the team and with this status that effectively exempted them from scrutiny, the Soviet Union knew of the vast majority of high profile intelligence operations against them before they could even leave the planning board. A good example of this is Operation Gold, a plan to dig a tunnel underneath East Berlin in order to tap regional Soviet military cables. But before the tunnel was even made, a man named George Blake compromised the operation, effectively opening up the plan and the agencies behind it to manipulation through misinformation. It is my view that the CIA was only kept in the game by strong counterintelligence agents like James Jesus Angleton and William King Harvey, but mostly Soviet walk-ins or defectors. Many of the CIA's counterintelligence victories were achieved through defectors like Walter Krivitsky and Michael Goleniewski.

In terms of counterintelligence programs, the early Soviet model was also far superior. Naturally, secret police, surveillance, and counterintelligence all come easily with a totalitarian, autocratic government. This enabled Red Army units like SMERSH (a name that amounts to "Death to spies" in Russian) to have free reign in convicting and killing spies as they were created as an external organ to existing Soviet military/intelligence structure. They were so effective because they targeted any threat to information security, both real and imaginary which resulted in the executions of many innocents.

So, as a brief answer to your question: during the early Cold War, the balance of power in intelligence (and counterintelligence) heavily favored the Soviets.

Sources: Vadim Birstein's SMERSH, David C. Martin's Wilderness of Mirrors, Michael Warner's The Rise and Fall of Intelligence

Related question - did the moles every uncover eachother? i.e. KGB mole within the CIA got told about the CIA's mole in the KGB and reported that?

Follow-up question: Is it fair to say that a Soviet agent would have an easier time integrating him or herself into US society and organizations? A recent immigrant from anywhere in Europe would be relatively unusual, depending on what part of the United States in question. However, wouldn't an American stick out like a sore thumb in Russia? Moreover, the (relatively) more open U.S. society seems like it would be easier to infiltrate. Even if we're talking about strictly the CIA/KGB, it seems that the CIA would be at least more ethnically diverse and open to application and promotion for "outsiders," whereas finding an Russian with an axe to grind would be more difficult.

Related question...was there any information gained by either the U.S. Or USSR that truly benefited either side during the Cold War? I'm not referring to information gained regarding the other side with relation to the intelligence community, but real world implications that would impact policy or the outcome of the Cold War? Or was it all just a spy vs spy game with two super powers essentially playing chess against one another?

Follow-up question: When I was growing up in the USA (80s/90s), my teachers made a big deal out of the knowledge that MI6 had been thoroughly and publicly compromised while the CIA was not. Some were open to the idea that it hadn't been uncovered, but that clearly the CIA was doing something better than the British and the implication was the Soviets weren't good enough to penetrate the CIA.

Was the CIA publicly known to have been compromised by then? Were my teachers victims of a propaganda machine? Or was the whole thing so clandestine that they really couldn't have any idea?

Follow-up question: Was there ever any evidence of spies as depicted in the show The Americans - where the spies are sent in young and they essentially live out their lives in the US pretending to be Americans.

In the theme of the question. What does AskHistorian's think of the book "Legacy of Ashes: The History of the CIA"?

Is there a comparative book on the Soviet espionage efforts during the same time frame?