Implemented by The initial design was published by Blockstream in 2014, but the implementation is blocked by the lack of native support for SPV proofs in Bitcoin (which may not be added at all). Rootstock workaround this by sacrificing decentralization (still work in progress). The Ardor platform created by Jelurida is the first to propose and implement the concept of Child Chains. Already running on testnet, the production Ardor launch is scheduled for Q4 2017.
A user on the parent chain first has to send their coins to an output address, where the coins become locked so the user is unable to spend them elsewhere. Once the transaction has been completed, a confirmation is communicated across the chains followed by a waiting period for extra security. After the waiting period, the equivalent number of coins is released on the sidechain, allowing the user to access and spend them there. The reverse happens when moving back from a sidechain to the main chain.
The sole distinction between public and private blockchain is related to who is allowed to participate in the network, execute the consensus protocol and maintain the shared ledger. A public blockchain network is completely open and anyone can join and participate in the network. The network typically has an incentivizing mechanism to encourage more participants to join the network. Bitcoin is one of the largest public blockchain networks in production today.