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.
“Further, contribution is weighted by computational power rather than one threshold signature contribution per party, which allows anonymous membership without risk of a Sybil attack (when one party joins many times and has disproportionate input into the signature). For this reason, the DMMS has also been described as a solution to the Byzantine Generals Problem[AJK05].”

It doesn’t matter if you’re moving $1bn or 0.01c across the Bitcoin network, you get the same security guarantees.   And you pay for this in fees and time.   What if you were prepared to trade safety for speed?   Today, your only real option is to send the coins to a centralized wallet provider, whom you must trust not to lose or steal your coins. You can then do all the transactions you like on their books, with their other customers and you never need touch the Bitcoin blockchain. But now you lose all the benefits of a decentralized value-transfer network.

Instant Payments: Since the creation of Bitcoin there has been a race for faster transaction confirmations. Instant payments allow new use cases, such as retail store payments, and transactions in online games. RSK carefully chosen parameters and new theoretical protocols (such as DECOR+GHOST) allow creating blocks at 10 seconds average interval, with low stale block rate, and no additional centralization incentives.
Things get a bit more interesting when you replace the single custodian with a federation of notaries by way of a multisignature address. In this model, a federation of entities must sign-off on movements to and from the sidechain, so more parties must be compromised for a failure situation to unfold where the bitcoins frozen on the main chain are stolen.
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Permissioned blockchains use an access control layer to govern who has access to the network.[46] In contrast to public blockchain networks, validators on private blockchain networks are vetted by the network owner. They do not rely on anonymous nodes to validate transactions nor do they benefit from the network effect.[47][better source needed] Permissioned blockchains can also go by the name of 'consortium' or 'hybrid' blockchains.[48]
The ShipChain platform unifies shipment tracking on the Ethereum blockchain, using a sidechain to track individual encrypted geographic waypoints across each smart contract. With this system, the meaning of each cryptographic waypoint is only accessible for interpretation by the parties involved in the shipment itself. This gives shippers more visibility across their supply chain, and allows carriers to communicate with ease.
"I see quite a few use cases for private blockchains, and they definitely have their place. Traditional institutions won't switch to a completely public blockchain from one day to the other. A private blockchain is a great first step towards a more cryptographic future. The biggest advantages of private blockchains in comparison to centralized databases are the cryptographic auditing and known identities. Nobody can tamper with the data, and mistakes can be traced back. In comparison to a public blockchain it is much faster, cheaper and respects the company's privacy. As a conclusion, it's better to rely on a private blockchain than no cryptographic system at all. It has merits and pushes the blockchain terminology into the corporate world, making truly public blockchains a bit more likely for the future." 
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.