The forever problem can be, ironically, resolved with crypto. If you encrypt each employer record with an employee key then, if the employee data needs to be deleted you just delete the key and the data is gone.
One problem with sharing data between multiple parties is that you need somewhere central to store it, somewhere that's ultimately trusted. If you decide to trust one party, they could still make mistakes, have downtime, stop supporting the project that requires the shared data, or go rogue.
I'm perhaps too ignorant of blockchain, but could it help with this scenario? Does it give a better promise of availability and more trust in its consistency and integrity?
edit: I'm thinking here of a blockchain whose only users are this group of companies, not a public system. I don't know if a blockchain becomes useless when it is only used by a small group.