Nothing.
There are no analytics frameworks, not even debug telemetry. That means a couple of things:
- I do not know anything about the folks using the apps aside from the App Store's download metrics or reviews left on the App Store page.
- If something breaks on your device, I'll have no idea that it's broken unless you tell me.
The only external calls that the app makes are to the official GW2 API using your API token. Those calls (along with your token) are stored in the app storage (or your browser, depending on platform) and are not sent anywhere else.
So, in short, we know nothing about you, not even when stuff goes sideways.