What is local Bitcoin?
LocalBitcoins is a service that allows you to trade BTC with real money. The difference between it and other exchanges is that LocalBitcoins staff themselves don't buy or sell, but its users set their trading fee, price, and the currency, and to buy or sell bitcoins the site allows you to find a merchant-user.

A local Bitcoin does not exist. The BTC blockchain is a whole and only meaningful if it's together.