The DISH Anywhere app will pull the Live TV and recordings over the Internet connection. Even on the same network, the Hopper will send the signal over the Internet to DISH Anywhere and the app will retrieve that signal over its connection to the Internet.
Sorry about "hijacking" the thread, but the subject matter is Dish Anywhere and more people thinking about it may get your answer faster.
Anyway, I Googled a general query "does dish anywhere use data?" and came up with the following thread from 'communities.dish.com':
DishAnywhere Bandwidth consumption on home network - mydish communities - 13284
It is four pages of much confusion and different answers just like this thread. However, there is one post that redirects to an answer from Dish employee Jason_C.:
Solved: Re: Dish Anywhere App and Local WiFi - mydish communities - 19012
He definitively answers that Dish Anywhere connects to a Dish server
first, then sends it, via Internet, to your device. As I said in my initial post, what got me to thinking about this was the (almost) doubling amount of data on my Xfinity home account. There are just three adults, so most data is/was used to watch Netflix, Amazon Prime, Youtube etc. and downloading the occasional PlayStation 3 game. But after making a home office out of a spare bedroom (that doesn't have a cable connection with both Hopper and Firestick on the same home network) is when the data usage increased. So I was down to
a) a neighbor was tapping into my home network or
b) Dish Anywhere was going outside and coming back on the Internet.
My experience and Jason_C tells me the latter is true.