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These phones will work without power because your voice acts as the source of current in the phone circuit. Because of the nature of these phones, each receiver is also transmitter; this means you can talk into the earpiece if your mouthpiece is broken and vice versa. When in a vacuum, remember that sound does not travel, so press your helmet against the mouthpiece when speaking.
Because of the relatively simplistic design of this device, a variety of nations have these onboard their starships. The Star Army of Yamatai uses Sound-Powered Telephones as a part of their Standard Emergency Systems Package. The Abwehrans use this device as a standard internal communications system onboard both military and civilian vessels allowing one less system to be powered by their power generation systems.
See this page for more (real-world) information about sound powered telephones.