Dual federalism is the political theory that two different governments share sovereign power over a certain region or people. Generally this is the concept of balancing the scales of power between a large, sweeping government and a more local, centralized one. Usually, this involves some sort of federal authority and a state regime. The theory works as a type of check to ensure corruption does not impact either government.
Rather than a form of cooperation, dual federalism is designed to create tension between the two forces. While the federal government mandates certain rules that cover a multitude of smaller governments, the state or local authority is responsible for nearly everything else. In most systems, the states are allowed to check the national authority through representation in the central government. Some countries even have a system of judicial oversight in which legal recourse is available to the state or local power.
Generally, a dual federalism maintains specific parameters by which the balance of power is maintained. National governments are only able to establish power through a system of laws. The overall purpose of the federal authority is limited by constitutional mandate. Throughout each government's sphere of influence, each authority maintains sovereignty that generally should not impact one another. In addition, it's important for both the federal and state governments to work together, but also maintain a certain level of distrust in order to operate efficiently and provide the best for its citizens.
Historically, the definitive example of dual federalism is the United States. The federal government is mandated by the US Constitution to maintain a series of laws defined by the Bill of Rights, constitutional amendments and US Code. In order to maintain control, representatives of the states are elected to a legislative body which creates these laws. The states themselves also have the formation of a government, which is responsible for all other rules. These states can check the federal government through judicial action.
Europe, too, has a system of dual federalism, albeit set up with state traditions. The European Union (EU) is organized into a federalist government with limited powers. Each member nation has its own system of government, some using a federalist system while others using methods such as constitutional monarchy in the United Kingdom or a presidential republic in France. The EU itself, however, enforces a series of laws but can be overridden by member states.