What is a blockchain, really?
At the fundamental level a blockchain is a way of receiving a community consensus about state changes in a (usually) pseudonymous fashion.
A block chain as it's name suggests is a chain of blocks, each of which contain a series of state changes and proofs that chain on from the previous block in such a way that the entire chain can be verified from a given genesis point all the way through to the current head block.
Block 1 & proof
Block 2 & proof
Block 3 & proof
In a proof of work system, the different miners are competing to provide the integrity of the system by solving difficult cryptographical problems. Integrity and auditability are ensured by making the blockchain public and paying participants to keep the system honest.
In a proof of stake system, the miner that gets chosen to mine the block is based off a slightly different algorithm to give the miners who have a bigger stake in the system a less difficult problem to solve than others with lesser stakes.
Private blockchains are almost a contradiction in terms unless there a multiple separate participants with separate concerns or interests keeping the other participants honest.
If there's a private blockchain with a single interested party controlling it then it's an almost pointless excercise, as there is nobody to ensure the integrity of the system. They might as well simply use a well protected database.
Public private chains
This is not the same as a public blockchain for a private (or specific) use.
If the general public (or enough different participants with cross purposes) can participate in the mining and ensure the legitimacy of the chain then that would allow integrity to be maintained.
You would however have to pay these participants in some fashion to encourage their engagement and honesty in the activity.
What they get paid (or their stake in maintaining the system's honesty) will determine their motivation to not be corrupted or bribed by other participants or actors in the system.
Generally you want to pay them enough to make them self interested in ensuring the system is not abused, to guard against the abuse of others and to make sure that the cost of bribing enough people to gain consensus is prohibitively high.