If I'm understanding the question correctly, there shouldn't be any inherent difference between a private and public network. Depending on how you are selling the tokens, you could either have the price adjust using logic in the contract, or off-chain.
On chain has the potential to be more transparent, as nodes on the network can see the contract logic, and can verify that the contract does what it claims to do. How transparent this is will depend on the setup of the private network. There are various on-chain price adjustment mechanisms you could looks at, such as Uniswap, Bancor, or KyberNetwork. Please note that for this to work, you would have to have a setup where means of payment for your token are on-chain, otherwise they don't have a means of interacting with the contract.
Alternatively you could set up an off-chain backend that receives payments, and when it detects payment, sends token to an address on your private network (maybe through a form).