Anyone who wanted to. If you wanted to make sure your personal data remained available when you turned your computer off or that it could be downloaded quickly then you'd need to arrange for paid hosting. Pick up cheap vps and share it with your family and friends. Whatever.
> how would that hosting mechanism be resistant to collusion and manipulation
It doesn't need to be, that's the point of using strong cryptography.
> it seems you will need some sort of decentralized state/data tracking (like a distributed blockchain ledger)
No, you just need to keep track of keys. Your identity is your key. We already do this. Certificate authorities, ssh keys, etc. The issue you're describing is one of using a centralized authority to solve the problem which is hardly the only solution and certainly doesn't require a blockchain.