A chain can be public or private in many ways – completely private where only certain parties can connect, or only private in that sense that certain parties are responsible for administration and "mining" (i.e. building blocks). There are anyone-can-* parameters to control all this: https://www.multichain.com/developers/blockchain-parameters/
If "mining" is based on permissions then it does not requiring "mining" in the sense of proof-of-work - instead, simple signatures are used with a round-robin consensus scheme. If mining is open to all parties then proof-of-work is used.
