We do sync the data to our servers, which you can switch off. We've engineered it so that if you just want to keep the data on your phone, the app will still work fine, and you can rely on your iTunes/iCloud backups.
Any plans for a kind of "export"? I like this idea a lot, as someone with a terrible memory but that doesn't use social media like Foursquare. I would only worry about a way to have my data permanently. Outside of the app which could be pulled from the AppStore at any time.
Engineer @heyday here. We've already got an export-to-json prototype, but it's not quite ready for the public yet. And of course we want to export into a beautiful format for the average user as well.
I don't consider being on my phone to really be an archive. People change phones, etc. Looks awesome but I wouldn't seriously use it if I couldn't export the data--in whatever degraded form--to a standard format. Frankly some sort of PDF export would be fine.
I basically can't use any journal app either unless I can export and own the data - as in, remix it into a different format if I choose. I've used two journal apps so far and have lost journaling data both times (both apps still exist). If there's a journaling app that lets me export to text/metadata every week to store in some repo, but where I still prefer to use that app for input and display purposes... that's a great journaling app.