There's an image of the Book of Honor in this publication (PDF - pg. 23) produced by the CIA that records CIA employees killed in the line of service. Covering the years 1950-1989, there were 66 employees killed. That's worldwide, though. And it probably doesn't include those who were indirectly employed.

In the United States of America there was only ever one Soviet Spy that was killed and it is quite possible and even likely that he killed himself.

These "illegal" soviet spies did nothing. According to KGB Maj. Gen. Oleg Kalugin they were completely ineffective and only ever managed to send open source documents which anyone with an internet access or library card could have obtained.

Many of them were actually caught and tried, but because of the fact that the never sent classified documents they were not convicted of espionage.

According to Oleg Kalguin when one was found out by the American spy agencies they would LEGALLY arrest them and then replace them with soviet spies that had either defected or were flipped.

According to Steve Coll in his book "Ghost Wars" he said that by the 1980's there was an unwritten rule that neither side would kill anyone under the direct pay and service of the other countries. This wasn't because the countries were less antagonistic towards each other during this period, but because the point of the cold war was to gather intelligence and killing each other constantly would get in the way of learning about each other.

There are no actual numbers though as it is still classified and will be, likely for the rest of our lives, but I can assure you that the numbers are very much so lower than what you see in the television show.

Sources

The First Directorate: My 32 Years in Intelligence and Espionage Against the West by Oleg Kalugin and Fen Montaigne

Steve Coll, "ghost wars"

Find, Fix, Finish: Inside the Counterterrorism Campaigns that Killed bin Laden and Devastated Al Qaeda (this one doesn't directly talk about the cold war, but it talks about the tactics and strategies of an intelligence war, which is what the cold war was)

Can I post a follow-up question?

In the show, in the aftermath of the Reagan shooting, several KGB personnel are shown to believe that some sort of coup were going on/possible to happen, further encouraged by Alexander Haig's controversial comment that he was (briefly) in control of the White House until the Vice President arrived.

Was this perception actually held by some of the soviets?

Granted, in the show, they only show a relatively low-level employee and the two agents who are completely and intentionally cut off from most intel.

Follow up question/semi related:

Sasha, the KGB source in the CIA. was he just the most notorious one, or were there others?

Forgive me, but isn't this asking how many secret operatives were killed by secret operatives, during a war that wasn't a war?