I do think it's worth thinking through privacy/security/freedom implications, here is my opinion (not necessarily my employer's):
All web apps can of course operate completely in offline mode (as well as privileged FirefoxOS apps[1], which have access to APIs to use more of the device features and are the bigger concern here) - all of the apps that ship with the phone do operate and store data offline, for instance (e.g. you can have a local calender, it's not stored on Mozilla's servers).
Knowing whether you can trust an app to treat your data as they say they will is not anything unique to the platform (who/where do you get it from, etc). Mozilla is very serious about this type of thing (I work full-time on various Mozilla server projects so I constantly experience first-hand how important this is to everyone, e.g. application design, handling crash reports, etc.)
However there does need to be a balance between the needs/desires of the platform provider (including built-in apps), third-party app developers and the end user - developers of third-party apps should have the ability to make choices about where data is stored, privacy policies and so on, and users should be able to install the apps that they want to.
[1] - https://developer.mozilla.org/en-US/docs/Mozilla/Firefox_OS/...