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How does a private blockchain such as VaultOS protect from a 51% attack? Is it somehow piggybacking on the public blockchain, or (I assume) are they running separate servers they control?
It seems if they are running their own hardware, that it is distributed but not protected by a huge volume of independently-run servers. I imagine they would run the servers independently but it's still a central authority running them so that has its limits.
Would someone please educate me?
The publicly available information about VaultOS appears to consist mainly of a series of buzzwords, and although that list has "blockchain" in it among other things it's hard to tell what the product is if they have one. But I'll answer this about systems like ErisDB and Hydrachain, which are actual, shipped, open-source pieces of software.
The normal idea with a private chain is that rather than having arbitrary people getting the ability to validate blocks by finding proof-of-work hashes, the validators are known to each other and identified by key-pairs. This means you know who the validators are so you don't have to worry about arbitrary people renting CPU power and taking over the system. Also since you know how many signatures should be required to sign off on a block, each participant can establish that you've got a [super-]majority of them agreeing to any given block, and accept that block irrevocably, rather than allowing a future 51% to come back later and rewrite history.
It's possible to make your own private chain using the stock Ethereum software with proof-of-work, and only open it to known participants. However beyond prototyping this generally isn't very useful, as your private chain is likely to have a very low difficulty, so any participant could rent a load of CPU power and take over the chain.
Private blockchain by definition is private (permissioned). You give a permission to access the blockchain only parties you know (consortium). You know the identity of these parties. There is no technical protection, but social control. If any of the parties misbehave they are kicked out of the consortium.
