Depends on the code, depends on the mechanism being used and so forth.

Consider a "secret message" type of code where words are broadcast over an open channel but there are specific code words that have secret meanings. For example, the British broadcast radio messages into France during the war which contained coded messages using special phrases as part of "Radio Londres". The intent being to broadcast news or information necessary to coordinate timing to those in the French Resistance. Prior to the D-Day invasion Radio Londres broadcast the first stanza of "Chanson d'automne" by Verlaine, which signaled that the invasion would occur within 24 hours and those in the Resistance with jobs to do related to the invasion should take action accordingly.

The bulk of coded messages were cryptographically encoded and sent via morse code over wire or broadcast. This would include messages encoded using the German Enigma machines or the Japanese Type B (US code name "Purple") code machines, among others. This made a lot of sense operationally because there was already vast infrastructure (both human and equipment/infrastructure wise) for morse code based telegraphy. To send or receive a coded message you simply insert an extra processing step in the operation. To encode a message using Enigma, for example, you write up the clear-text message, then format it, you set the Enigma machine (which is basically just a fancy typewriter) with the appropriate code wheel configuration and then you type in the clear-text which produces an enciphered text. Then you hand the enciphered text to a radio operator who reads the letters and bangs them out in morse code over the radio. While on the receiving end someone reads the morse code and transcribes the message into a recorded text (either with a typewriter or other kind of stenographic machine or by hand using a pen or pencil on paper). A process which had been well established for decades prior and which had no shortage of individuals capable of rapidly sending or receiving messages in morse. Then the decryption is just another step using an Enigma machine that's properly configured which recovers the plain-text from the enciphered text. A common pattern was for ciphertext to be formatted in five character blocks organized in columns since otherwise they would simply be a long unbroken mega-paragraph with no punctuation or white space which humans aren't great at dealing with.

Spying on these messages was mostly a process of listening in on the transmission in real-time and having someone record the text of the enciphered messages sent in morse. Tape or wire recording was possible but very rare at the time.

There were also some "secure speech" systems used during the war though they were generally reserved for higher level communications. For example, the A-3 system used by the allies was sort of a scrambler/de-scrambler system that the Germans were able to crack through spectral analysis. A successor system called SIGSALY used semi-digitization, bandwidth compression, and pre-recorded white noise samples on vinyl records to successfully encrypt phone calls between high-level officials. If you listened in on these transmissions it wouldn't quite sound like pure white noise but it would sound very electronic with no discernable speech patterns. This system required multiple cabinets of electronic equipment on each end along with copies of the noise records used as well which had to be played in synchrony.

Messages enciphered by Enigma were transmitted in Morse code, when intercepted they were generally transcribed by hand. In Britain interception was the domain of the Y Service, its operators would complete W/T Red Forms as they monitored transmissions with the raw intercepts despatched to Bletchley Park for deciphering.

In 1941 the Y Service started picking up non-Morse transmissions; these turned out to be radio teletype code used by more advanced cipher machines such as the Lorenz SZ40. These could not be transcribed by hand, so were recorded using undulators. The resulting visual representation could then be turned into Baudot code by 'slip readers'.

For more on the Y Service and its worldwide operations there's Kenneth Macksey's The Searchers: Radio Intercept in Two World Wars or Sinclair McKay's The Secret Listeners: How the Wartime Y Service Intercepted the Secret German Codes for Bletchley Park; on the teletype codes there's an excellent section at the start of Paul Gannon's Colossus: Bletchley Park's Greatest Secret.