Glossary
On-device speech recognition
Speech-to-text performed by a model running on the device, not on a vendor's server.
Speech recognition turns audio into text. On-device recognition does that with a model stored and executed on the computer. The alternative is a service that receives the audio and returns text. The distinction matters because the audio, for the recognition step, stays put.
Murmur uses a local speech model. Hold a hotkey, speak, and the text is typed into the focused app. That claim covers transcription. It does not cover what you do with the text afterward. Sending an email sends the text through the mail app. The longer note is What stays on your Mac when you dictate.