They released the source (well, currently only the Android version) at https://github.com/google-ai-edge/gallery .

At a glance, I see they do gather analytics about how much the app is used (model downloads, model invocations etc) without message content, pretty much just the model used.

> ...your shoe size

The funny thing is that a lot of Google's internal training content uses an imaginary product "gShoe", and discusses the privacy implications of data that such a shoe might collect :D

Apple is paying Google $1billion for an AI strategy that runs on device. We're seeing the preview of what that will look like.