Mining is a heart of Proof Of Work consensus algorithm. It is essentially a puzzle which miners have to solve in order to generate proof of work and validate a transaction. It is computationally intensive.
Private blockchain has different consensus algorithms for validating transactions.
For example,Quorum is popularly used for developing enterprise blockchain applications. It uses voting based consensus algorithms such as Raft, Quorumchain, and Istanbul. geth has an option to use Proof Of Authority consensus algorithm for private blockchain network.
In all the above-mentioned consensus algorithms process of validating blocks remains the same i.e. solve the cryptographic puzzle(mining).
In the case of Raft consensus only elected leader will do mining and broadcast it to all the participants in the blockchain network.
In case of Proof Of Authority consensus algorithm of geth only authorized nodes will do mining.
Proces of validating block is the same(mining) depending on consensus algorithm that you select for private blockchain only selected nodes will do the process of mining. In the case of the public blockchain network, all the participating nodes will compete for rewards received after mining the block successfully.
