For the iOS app it connects directly to the TiVo or stream through your WiFi and does not stream over the internet. It stays on your local network. I believe that the Android app works the same way.
What is unclear to me is what happens when you stream to a PC on your local network using online.tivo.com. Obviously it uses the tivo servers on the internet to connect to your tivo/stream, but once you start streaming I don't know if the streams goes from your local tivo to the tivo servers and back down to your PC all over the internet connection, of if once you start streaming it goes directly from the tivo to the PC and stays only on your home network.
Hopefully someone else knows what happens in that case.