Does anyone monitor/support public-sector scientists?
Your questions appear to be asking about non-academic government scientists. 

To answer your question about supporting science, yes groups exist that support them. The largest of which is probably the American Association for the Advancement of Science (AAAS) who advocates on behalf of scientists both in the United States of America and other countries as well. The mission and purpose of AAAS is to 
advance science, engineering, and innovation throughout the world for the benefit of all people.

I am member of AAAS and a government research scientist. The news and updates section of Science includes both news about government scientists and what AAAS is advocating on behalf scientist and science. 

To answer your question about monitoring science (at least in the United States), the Office for Personnel Management (OPM) provides oversight and standards for managing scientists across agencies (e.g., this public webpage lists some of their resources).