There are 3 scopes as you might already know:
1) Default Scope (or Site Scope or Community Scope)
2) Page Scope: The scope of the page. This can be considered as a sub-scope of Site Scope.
3) Global Scope: This is used to show the content irrespective of any Site or Organization as you have thought. It is available to all the sites or Organization Sites.
Here is how you can configure any portlet on a page to show the Contents of the Global Scope in that portlet:
Click on Configuration --> Scope --> Select Global Scope --> Click Save [see image attached]
