Twilio developer evangelist here.
It sounds like you are using Twilio Video (since you mention dominant speaker events). Twilio Video itself doesn't have "audio speaking" events, neither does the web platform itself.
You can however do some audio analysis in the browser to tell whether a person is making noise and you can compare that to whether their audio track is currently enabled in order to show a warning that they are speaking while muted.
To do so, you would need to access the localParticipant's audio track. From that you can get the underlying mediaStreamTrack, turn it into a MediaStream and then pass it to the web audio API for analysis. I have an example of doing this to show the volume of localParticipant's audio here: https://github.com/philnash/phism/blob/main/client/src/lib/volume-meter.js.
Once you have that volume you can then choose a threshold where you decide a user is trying to speak and then compare whether that threshold is broken while the user is muted.
Let me know if that helps.
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