The bridge between multiple blockchains are the private and public key pairs. The keys are what enable you to sign transactions. In the case of Ethereum blockchains, the same private key can sign a transaction on a private blockchain as well as a public blockchain.
In the Azure Quickstart Template example, the address
7fbe93bc104ac4bcae5d643fd3747e1866f1ece4 is allocated lots of ether in the genesis block. In fact, it is the only address which has ether allocated to it. The
priv_genesis.key file that comes with the tutorial contains the private key associated with the address. The tutorial has you import the private key and then sign transactions with it.
On the public Ethereum network, if you were to send ether to the address
7fbe93bc104ac4bcae5d643fd3747e1866f1ece4 then you could sign transactions with the exact same PK found in
priv_genesis.key. As the walk through mentions, you do not want send ether to the address
7fbe93bc104ac4bcae5d643fd3747e1866f1ece4 on the public Ethereum network since this private key is exposed to the world, anyone could take control of those funds. As you can see, no one has ever sent funds to this address, as block explorers can't find them: https://etherchain.org/account/7fbe93bc104ac4bcae5d643fd3747e1866f1ece4
Being able to send transactions between two separate blockchain networks (blockchain interoperability) is a much more complicated discussion. There are efforts such as BTC-Relay (http://btcrelay.org/) which are making incredible strides to solve pegging between blockchains.