I noticed that if I turn on Do Not Disturb on the watch, it turns on on my phone and vice versa.
The same thing happens with Airplane mode.
However I noticed a thread on MacRumors saying that turning the phone's hardware mute switch to silent should also turn on mute on the watch. But this doesn't happen with mine.
Even with my phone on hardware mute, the watches sound is independently set, and doesn't turn on mute on the watch.