Bulbs and switches use ZigBee ZLL (AFAIK, they don't have WiFi capabilities at all), and IIRC it can even work without the bridge (just turning on and off) as switches can talk directly to bulbs.

Bridge device surely can work without Internet. It's essentially a small webserver that exposes an API for the smartphone app (which talks to the bridge directly as long as they're on the same network) and then talks to the devices over ZigBee. IIRC, there is even FLOSS RPi-based alternative implementation somewhere out there.

Don't know if Internet is required for the initial setup.