The ledger is already distributed. All that is left to do is connect it.
Much of our thinking around distributed ledger technology (DLT) is backward. Personal information should be stored locally on personal hardware devices, and served to those who request it, not stored in centralized databases under an associated, yet disparate, user id.
Public blockchains, like one for land or car registry, for instance, or any other transaction for that matter, should be the repository for the associations and outcomes of said transaction. For example, a hash on a public blockchain may reflect that KEY XYZ transferred TOKEN 123 to KEY ABC. How much about KEY XYZ, TOKEN 123, and KEY ABC is made available is up to those who have a stake in the transaction choose to reveal.
All this not only already exists but also occurs in both our digital and real life. The missing link is connecting the distributed ledger of our personal information with the public ledger of our social interactions with other entities.