No, it's my device that creates the data, so it is mine. Otherwise, does the typewriter manufacturer own the stories you type?
Here's an example: You go to the library (or a privately owned web cafe), and write your stories there. It doesn't matter that you don't own the device. It's still your data.
Once money exchanged hands, it's not their product, it's mine, as is _all_ of its output, including logs, etc.
Otherwise, I'd call it 'spyware'.