How can the nodes on one Ethereum private blockchain send/receive transactions from the nodes operating on another private blockchain?
By design, nodes talk each other only if they are on the same Ethereum blockchain (same networkID and genensis block). Maybe you can achieve something like a two-way peg sidechain with two Ethereum blockchain, for example using some adapted relay contracts like BTC Relay.
For completeness, while nodes can't cross their own blockchain borders, Ethereum transactions are not bounded to a specific blockchain at all, so you can actually send the exactly same signed transaction to different blockchains using different client nodes. Indeed, this is why you can send for example a transaction both to the ETH and to the ETC blockchain.