The bridge will still work from your private network and one can still control the lights with the app.
All that's being disabled is the external control.
All that's being disabled is the external control.
If you're willing to set up port forwarding (and DDNS if needed) there are apps that allow you to enter both internal and external addresses, so you can still have external control.
I have a V2 with "Out of home control" set to disabled and it works fine with my Echo Dot and with the Amazon Alexa apps on my phone and iPad.
My guess is that the bridge sends status information, including the bridge's IP address, to the Hue site, and the Echo can retrieve that information in order to find the IP address to send commands to.