You are looking at it wrong. Transactions are signed with privatekeys. Same adresses have same privatekeys on both networks. Mpcoin nodes don't have to communicate with Gavincoin nodes to doublicate the transactions.
There are not two networks. There are two sets of nodes which disagree on which is the "One True Blockchain". Up until the fork (a certain block height) all the nodes have 100% agreement, but then after the fork, a subset, "MPnodes", start to reference utxo from blocks which are
unknown to the Bitcoin nodes. These tx are rejected by Bitcoin nodes just the same way, as if I create and broadcast a tx for 123,000 BTC to send to an address I own, so I can have a nice fat bitcoin holding to rival the Winklevoss'.
Companies which accept bitcoins for products and services will be anxious to ensure that their nodes conform to the majority view and have the same blockchain that the exchanges, and payment processors do.