The clicking that was heard by the parties during the CW signaling happened when the called party got service from an AT&T 1AESS, which is an electronic switch, but is analog. It was the first generation of electronic switches, and there are still a few in service today. The switch serving much of Oakland Main was a 1A until around 2002-03.
In those switches, the line receiving the call had to be physically switched out of the conversation, connected to the bus with the 440 Hz tone, then physically switched back to the conversation. That meant the called party heard <click><400 Hz tone><click>, and the other party heard <click><nothing><click> at the same time, since on those switches, one heard a click for connections/disconnections.
The majority of AT&T subscribers in CA now are on DMS100s or AT&T 5ESSes, which are fully digital switches. (Oakland Main is a 5E.) In those switches, the voice path to the line with CW is momentarily replaced by the 440 Hz tone. The called party just hears the tone (similar to my experience with the subsequent beep on my Fusion). The other party may notice a momentary dropout of the voice or background noise of the called party.