Which area did the United States' state and federal governments share power?

United States' state and federal governments share power in mainly collecting taxes.

Explanation:

There are a few powers which are delegated only to the federal government and few are reserved for the State. Like, States operate their own judicial system, regulate property rights etc.

But, there are many powers which belong to the federal government are shared by state governments. These powers are known as 'concurrent powers'. These are powers like collecting taxes, spending or borrowing money the power to tax, spend, and borrow money. These also share powers like making roads, protecting environment, regulate elections,.

State governments operate their own judicial systems, give public education, and regulate property rights.