There is a kernel of truth to this story, but the full story is much more remarkable than I expected.
- Kagoshima-ben was used on phone calls from Japan to Germany when coded telegrams failed.
- Unbeknownst to the Japanese, the intercepted messages were successfully deciphered by a Japanese-American who had spent time in Kagoshima.
- This Japanese-American was a remarkable person. As he worked as a translator, his wife and children were imprisoned in an internment camp. But he defended internment.
- Largely as a result of this, his legacy has been neglected in the United States. And he has become mildly famous in Japan for the wrong reasons.
David Akira Itami (1911-1950) was a Kibei, meaning a second-generation Japanese-American who was sent to Japan to receive a suitable Japanese education, then returned to America. He happened to be sent to his parents' home in Kagoshima and received a well-connected education.
Kibei were an unusually politicized group among Japanese-Americans. They were a small group of about 11,000 people, but they ran their own Japanese-language press. Many of them were fascists, having been indoctrinated in militarist Japanese elementary schools. Those were not fascist were communist. Itami was neither. He had no quarter with traitors so went to the side of the leftists, but then alienated them by saying that the USSR was more dangerous than Japan, and finally led a Red Scare purge of the Japanese American Citizens League, which made him many enemies.
To directly answer the question, the Kagoshima dialect communications were intercepted by the US and recorded, and were indeed hard to translate. From the sole English-language article about Itami:
Akira, of course, had grown up with this dialect, and this made his work vital to the American forces. When, for example, the MIS intercepted a conversation in this dialect between Maki Hideji, head of the Broadcasting Section of the Foreign Ministry’s Intelligence Bureau, and a Japanese attaché in Berlin. Itami not only could translate the message; he knew the Japanese diplomat personally. It was Sogi Takateru, Itami’s mentor in Kagoshima, who had procured financial aid for him so that he could return to California in 1931. The message was an important one: it consisted of a report from the European Theater and a discussion about the strategic positioning of the Japanese fleet in the Pacific.
(Additional sources in Japanese here.)
Later, Itami was the lead interpreter at the Tokyo Trials. A few years after that, he killed himself. Having attacked fascists and communists and defended the internment camps, and yet being suspicious as a foreigner closely linked to Japan, he was friendless among Japanese, Americans, and Japanese-Americans alike.
In 1983 the novelist Yamasaki Toyoko became interested in Itami's life and produced a work of historical fiction based on deep research. Yamasaki portrayed Itami as loyal to both nations, and posited that he committed suicide because he was ashamed by the victor's justice at the Tokyo Trials, and felt guilt for having assisted the kangaroo court. It doesn't seem likely that any of this was true -- it reads like projection.
Itami's troubled legacy didn't end there. Thrilled at the depiction of Itami as a patriot of two nations, Japan's public broadcaster produced a TV series about his life in 1984. But this film was blocked from screening in America by the Japanese-American community, which was busy trying to obtain apology and reparations for internment. Itami was now not only misunderstood in Japan but also inconvenient to America, more than three decades after his death!
For this reason he has been totally forgotten in the US, and his fame in Japan is mixed up with a weird depiction of war... by request, here is a clip from the 2019 film fictionalization of his life.