Your emails are already at 'home' or a similar cloud solution with questionable privacy policies. These assistants don't actually run on your device, just their front ends are run on your device. I think you're making a distinction that doesn't really exist with common use cases.

As far as what is reported back to the home company, well, that pre-dates personal assistants. What gmail does with your email or dropbox with your files and any analysis your usage is a completely separate issue than personal assistants, aside from both of them having to do with privacy. It seems to me these assistants are just dipping into stuff 'home' has had access to for a decade plus. Instead of using that info to sell to marketers or whatever, its using that info to provide value to you by powering your assistant.