Federal Reserve is responsible for fiscal policies and US Treasury responsible for Monetary policies. Right?

Question

Or else, Do each take the responsibilities of the other?

Answer 1

It's the other way around

Try to think of fiscal responsibilities as those that involve government spending on social services, running agencies, military, and any other spending that our elected officials have approved

Monetary responsibilities deal with the stability of the dollar (in the US) and have to do with the management of the Central banking system, which  is what the Federal Reserve is, the Fed.

The Fed does not deal in political affairs but is largely independent, while the US treasury carries out the spending habits of our government.