What do you mean by "local"?

If you mean "on my network, never going to the cloud, install whisper and piper apps (in default repos at that) in Home Assistant OS: you have a fully local voice stack that multiple assistants can use. If you might have multiple speakers, why duplicate resources for processing speech?

HA basically compartmentalizes voice into into its parts: speech to text, intent processing, and text to speech. It's fully local if you want, or cloud, or both, entirely up to you.

If you mean "this software processes voice itself from the attached microphone, interprets the command, sends the command to HA, and then creates a TTS response or chime", there are software for that but they certainly aren't running on an inexpensive ESP32-S3 chip, and are generally gonna be pretty limited as well, and I don't see active projects using it anymore. Everything that was just...tended to work poorly.

I believe I've seen some self-contained software packages that can connect to HA's Assist over an API, feeding it the transcribed text directly, but again, a central source of processing speech on more-powerful hardware that is only ever going to be used like, one user/speaker at a time makes way more sense than duplicating efforts on worse hardware.