OK, here are my test results:
1) You DO need WiFi to do screen mirroring (or for the Chromecast to work, period!) and you need to be in the same WiFi for it to work (I did try with two different routers and if on different WiFi networks, the Chromecast app, where you have the screen mirroring option, doesn't find the Chromecast).
2) You DON'T need Internet for Screen Mirroring to Work, so if you're in a hotel, you can set up your laptop to act as a hotspot and connect both Chromecast and phone to it, even if it doesn't have internet...and if it does, well, it'll just share it.