I've installed the android app, because well reasons, and through I have a somewhat inexpensive data plan (I normally have enough for a month), that doesn't mean that I leave it without control. So through it is elsewhere that the API sends gzipped data, I couldn't find anything about the iOS nor Android app.
Yes, data downloaded from our API is compressed. As pointed out in the API docs on compression:
During normal operation, we guarantee that all responses are compressed, either with GZIP or DEFLATE.
However, although many image assets that are downloaded by the app (e.g. site icons) are not compressed, they are cached on the device.