What open-sourced and accurate speech-to-text engines and APIs currently exist?
Instead of speech-to-text (STT), Wikipedia has “Speech recognition” , otherwise known as automatic speech recognition (ASR). Wikipedia also has a higher level “Category:Speech recognition” ; under which, you can find a “List of speech recognition software” . In terms of open source, Wikipedia includes an entry for “Speech recognition in Linux” .
Of course, Nuance is the industry leader. AT&T is also promoting their “Watson Voice Recognition Technology & Speech API” . Red Shift Company  offers “RASR Speech Recognizer”. Koemei API  offers speech to text for video transcription. Google also has their undocumented HTML5 Chrome speech API. And in Windows, there is the Microsoft Speech API .