Do we have any Android hackers here? Is it possible to somehow run an app in a sandbox to check exactly what information it accesses and sends over the internet?
I wouldn't call myself a hacker, but many custom roms have an app control panel that allows you to deny permissions to apps. It also tells you the last time a particular permission was used. A ton of apps out there ask for permissions for things that only get used if you use one very specific feature of the application, and otherwise it goes unused. For example, banking apps will often register for location access and the ability to make phone calls (the latter shows up as "Services that may cost you money") but they only get used if you search for nearby ATMs, or hit the call a branch button, respectively.
Personally, I doubt they have the bandwidth to track people down based on what websites they visit, unless they require you to give personally identifiable information directly into the app. That said, I can't think of a reason anyone on this board would want to install it outside of morbid curiosity.