There's the mute switch on the phone (I don't have an answer for it)
There's the mute toggle on the watch (tom504 covered it)
There's Do Not Disturb, which can be set to mirror your phone (Watch settings -> General -> Do Not Disturb ->)
Which version of Mute is the question about?
[edit]
Ah, ok -- the switch on the side of the phone.
I don't think there's a way to tie them together. I have my phone on silent almost all the time anyway (and with vibrations turned off completely), and my watch is almost always on silent, too.
I hate hearing other people's phone ringers, especially when it's ringing and buzzing and they take another ten seconds to look at the screen before doing anything about it despite being able to silence it with a single button press, the heartless bast....