I searched the forum and the question already came up but it does not seem to be answered:
Here I found the permissions needed and network does not seem to be one of them (but I am actually not sure if it was considered a permission):
It would be very nice of @mylesofsmiles to confirm that network is not needed and if so whether you could use the pixel watch without connecting it to wifi or cellular. So that in theory it has no possibility to connect to the internet at all.