So let's take this as a scenario:
I have shares in a company in Australia, The company decided to get into the US and do the same service but it'll target US customers.
Will the new company have a fresh shares pie? or do both companies share the same shareholders and their percentage of both companies.
You can operate a branch as a remote office but that has legal issues - or you open a separate company. In the later case the company will simply be owned by the parent company for 100%. Microsoft Germany (GMBH - german form of limited liability company) is owned by 100% from Microsoft Corporation USA and sends all profits there.
So, no new shares for the owners of the parent company, but more value in the shares of the parent.
It depends. You can have a company be a completely separate entity namely a subsidiary which has its own equity part or whole of which is owned by the parent company in Australia. Normally though the shares of the company are the same if not for nothing just to keep it simple.