Yes, absolutely. Both sides constantly spied on each other, and on their own people, throughout the Cold War - and they used a variety of techniques and technologies to do it.
The Soviets and the West had different styles of espionage. The Soviets preferred to utilise human intelligence - the old-fashioned method of using people as agents to gather information on your enemy - while the USA was far more reliant on technology.
This is chiefly because Soviet society was much more closed than Western society, which meant that it was very difficult to cultivate assets.
There were plenty of defectors, but there were comparatively few spies for the West embedded within the Soviet Union itself. Western society, on the other hand, was a much softer target for Soviet intelligence.
Researchers think that in the 1950s around 70% of the US's intelligence budget was being spent on developing technology, including things like spy satellites, spy planes, and even spy submarines.
The US's smaller-scale spy gear - gadgets - lagged behind this large-scale spy tech. Even by the early 1960s, while the US was spending billions of dollars on space age spy tech, their gadgets hadn't moved on that much from World War II.
This all changed in 1962, when the Soviet defector Oleg Penkovsky was caught spying for the West, tried, and executed by the KGB. His old-fashioned spy camera played a key role in his show trial, and it prompted the CIA’s Technical Services Division (later the Office of Technical Service) to develop more advanced, harder-to-detect gadgets.
These included obvious things like secret communication equipment and pen cameras. But there were a few more outlandish examples too, such as the failed Acoustic Kitty programme.
This involved performing surgery on cats to install microphones and radio transmitters into their heads for the purpose of eavesdropping on conversations in the Kremlin and Soviet embassies (this is not made up).
The Soviets did develop gadgets of their own, but weapons were more their style than spy gear. In 1961, the KGB officer Bohdan Strashynsky used a cyanide gun hidden inside a rolled-up newspaper to assassinate two Ukrainian dissidents.
In 1978, the Bulgarian dissident Georgi Markov was killed in London, reportedly (though not confirmed) by a gadget known as a "Bulgarian umbrella" - a normal-looking umbrella which fired ricin pellets.
There is also the famous "kiss of death", a small single-round pistol disguised as a lipstick holder, developed for the KGB's female spies. But how widely this was deployed isn't clear.
So, as you can see, some of the real-life technology was even more outlandish that the movie caricatures.