“A private blockchain is hardly different from a traditional database. The term is synonymous with glorified databases. But the advantage is that if they are to ever start adding public nodes to it then it becomes so much more. An open blockchain is the best method for having a trustless ledger. The broader the range of decentralized adoption the better. The Bitcoin blockchain hits all those points. 
@mowliv I think a good way to think about it is by looking at our economy. The Federal Reserve prints US dollars for the US Government (the main blockchain) to boost the US economy. However, US dollars can be exported to other countries (a side chain) that could have a completely independent economy but still use a currency backed by the US government. – Olshansk May 30 '17 at 0:56
Blockstream believes that to be secure, blockchain systems must be built with open source technology. Towards that goal, we've created the Elements Project, a community of people extending and improving the Bitcoin codebase. As open source, protocol-level technology, developers can use Elements to extend the functionality of Bitcoin and explore new applications of the blockchain. Join the expanding group of individual and corporate developers using Elements to build robust, advanced, and innovative blockchains.
The Loom Network recently released their SDK which supports what they call “Dappchains,” an Ethereum layer-2 sidechain solution with each sidechain comprised of their own DPoS consensus mechanism. This enables highly scalable dapps, specifically games built using their tools. Loom emphasizes the earlier comment about sidechains enabling innovation in scalability, rather than providing it directly. Loom’s sidechains have their own set of rules and are used to offload computation from the primary Ethereum chain. Their sidechains are application-specific, meaning that they enable highly scalable dapps through an efficient consensus mechanism and can periodically be settled on the main Ethereum chain depending on their security needs. You can find more information on their model here.
When you send Bitcoins somewhere, you lay down the challenge for the next owner. Usually, you’ll simply specify that they need to know the public and private keypair that correspond to the Bitcoin address the coins were sent to. But it can be more complicated than that. In the general case, you don’t even know who the next owner is… it’s just whoever can satisfy the condition.
These kinds of blockchains are forks of the original implementations but deployed in a permissioned manner. Mainly hyped because the companies behind these chains want to onboard corporations in order to generate buzz around their their chain. It’s tolerable for proof of concepts or if they plan to move to public as soon as possible; otherwise they are just using the wrong set of tools for the job.

Public blockchains provide a way to protect the users of an application from the developers, establishing that there are certain things that even the developers of an application have no authority to do. From a naive standpoint, it may be hard to understand why an application developer would want to voluntarily give up power and hamstring themselves. However, more advanced economic analysis provides two reasons why, in Thomas Schelling's words, weakness can be a strength. First, if you explicitly make it harder or impossible for yourself to do certain things, then others will be more likely to trust you and engage in interactions with you, as they are confident that those things are less likely to happen to them. Second, if you personally are being coerced or pressured by another entity, then saying "I have no power to do this even if I wanted to" is an important bargaining chip, as it discourages that entity from trying to compel you to do it. A major category of pressure or coercion that application developers are at risk of is that by governments, so "censorship resistance" ties strongly into this kind of argument.
This type of permissioned blockchain model offers the ability to leverage more than 30 years of technical literature to realize significant benefits. Digital identity in particular, is fundamental for most industry use cases, be it handling supply chain challenges, disrupting the financial industry, or facilitating security-rich patient/provider data exchanges in healthcare. Only the entities participating in a particular transaction will have knowledge and access to it — other entities will have no access to it. Permissioned blockchains also permit a couple of orders of magnitude greater scalability in terms of transactional throughput.
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