The Ethereum subreddit has this post related to an Ethereum-Dogecoin sidechain.
What is a sidechain?
Could one sidechain be a private Ethereum network?
Put simply, sidechaining is any mechanism that allows tokens from one blockchain to be securely used within a completely separate blockchain but still moved back to the original chain if necessary. By convention the original chain is normally referred to as the "main chain", while any additional blockchains which allow users to transact within them in the tokens of the main chain are referred to as "sidechains". For example, a private Ethereum-based network that had a linkage allowing ether to be securely moved from the public Ethereum main chain onto it and back would be considered to be a sidechain of the public network.
Sidechain is a blockchain that runs in parallel to the main blockchain which extends functionality through interoperable blockchain networks allowing a decentralized way of transferring/synchronizing your tokens between the two chains. In other words, you can move your cryptocurrency to the sidechain and then back to the main chain.
Sidechains are decentralized, peer-to-peer networks that provide useful enhancements (such as security, risk, and performance) for global systems of value exchange that don’t need any other third parties. They enable developers to safely develop new applications without a risk.
They're very similar to Bitcoin sidechains. They tend to improve scalability, but only indirectly by enabling innovation.
Here is a wonderful explanation of state channels and sidechains, their working and how they differ from each other, and their pros and cons.
According to the article:
State channels are the general form of payment channels, applying the same idea to any kind of state-altering operation normally performed on a blockchain.
A sidechain is a separate blockchain that is attached to its parent blockchain(mainchain) using a two-way peg.
The two-way peg enables interchangeability of assets at a predetermined rate between the parent blockchain and the sidechain. The original blockchain is usually referred to as the ‘main chain’ and all additional blockchains are referred to as ‘sidechains’. The blockchain platform Ardor refers to its sidechains as ‘childchains’.
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.