The political idea / system of government that has separate and shared powers between the national government and the local regions is called ...

Federalism

Explanation:

Federalism is a system of government with two distinct power structure. It is divided between a National/Central government and Local/ Regional governments.

The authority of power is divided at dual level to ensure proper functioning of the government. The National government holds complete authority over the nation but it does not interfere in the powers bestowed upon local government. They both work independently but under the guidance of National government. Some extreme cases involve both of  these whereas some cases do not.

In short the division of power is an essential tool to create a balance and share of government's responsibility.

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