Provided you are using the same algorithm that ETH uses for your private chain, you will always have to mine in order to create blocks and process transactions.
Since it is a private blockchain, there should not be many miners on the network, and thus the hashing power should be low. Because of this, you are able to mine with a simple CPU and very little resources. The same mechanisms are working behind the scenes on your private blockchain as they do on the main Ethereum network.
Also is it require to have mining in private and permissioned blockchain?
This is up to you and your use case. If you want full control over transactions and hashing power, this is probably the way to go.
One option you have is to run a proof of authority network (PoA). This is a network in which the exact miners are defined, and only they are able to mine. Nobody else is allowed to contributed hashing power to this type of netowrk.