Which is precisely why the usefullness of private, permissioned blockchains is questionable. In the end they lose all the worthwhile properties of a blockchain -- including, effectively, immutability -- leading the concept of blockchains ad absurdum.
The upcoming data directive of the EU is merely yet another reminder why it makes no sense to try cramming blockchains all over the place. The blockchain use cases that "got lost" due to the upcoming EU directive were likely never viable in the first place.
Would make for an interesting legal case though. Say you're a company that offers the service to store your personal data on a public, permissionless blockchain by offering a simplified user interface for the likes of OP_RETURN. Who has the legal responsibility over the data? The consumer that made use of the service or the service provider?
Exactly.