-
Hue communicates locally without an internet connection.
The Hub does all of the work.
However the Hub may require an internet connection to be set up initially.
I do know from monitoring network traffic that once a Hue Hub is set up, that if you run the app on the same Local / Wi-Fi network that you can talk to the Hub and work the lights without the internet.
I'm unsure on the set-up though... because my internet was up when I configured it.
Note that this is why there's a whole different Philips Hue for "Commercial"... needs a different level of hardware and you can't just plug consumer things in and hope they work.