That's like... the whole point? You have some kind of hardware-based measured boot thing that can provide a cryptographic attestation that the code it's running is the same as the code that's been reviewed by an independent auditor. If the auditor confirms that the data isn't being stored, just processed and thrown away, that's almost as good as on-device compute for 99.999% of users. (On-device compute can also be backdoored, so you have to trust this even in the case that everything is local.)
The presentation was fairly detail-light so I don't know if this is actually what they're doing, but it's nice to see some effort in this direction.
E: I roughly agree with this comment (https://news.ycombinator.com/item?id=40638740) later down the thread -- what exactly the auditors are verifying is the key important bit.