When I think of telephone tapping equipment during the depression era, the first thing that comes to mind is "Ricco" from the old 50's TV program, "The Untouchables," standing on top of a telephone pole in the cold, pouring rain. Something close to then using alligator clips to attach his device to listen in; or something like that. Watching it as a kid, I used to wonder if "that poor guy had to sit up there until the case was over?"
(I did find it ironic in one document I found where Hoover and Purvis were concerned that the news media was tapping Chicago FBI lines and Purvis was instructed to go outside the office to another phone. Now there's a twist!)
Anyhow, for "the gaget person within you," this 1935 document reveals some of the apparent devices used at the time, and ordered by HQ, as early as 1933. Foote-Pierson, Vacuphone and other items are mentioned. I never looked into the history of LE sound equipment so if I saw one of these at a yard sale, I'd probably walk right by it thinking it some sort of early medical device for the "old timers."
I'll leave the history and research of this equipment to others but the 1935 document mentioning them is here.