Does anyone monitor/support public-sector scientists? The government science sector is relatively small, but when you add up all the state and national departments and related organisations, it's a lot of people doing a lot of science.

Despite the strategic importance of public sector science, as far as I know there is no group responsible for the sector as a whole. It is no one’s job to support, connect or speak out for the highly diverse, often committed but sometimes isolated workforce within the government science sector. It is no one’s job to keep track of the sector, to share knowledge and work towards practical resolution of shared issues. This stands in contrast to the relatively well-organised, represented and funded science in academia. (Note that public sector jobs can be well paid and secure)

I would like to know if there are any organisations, initiatives or resources aimed at a) monitoring or b) supporting government (public sector) scientists, in any country. 


One excellent example is the UK's Government Science and
Engineering profession. Well worth a look.
https://www.gov.uk/government/organisations/civil-service-government-science-engineering


What do I mean by government / public sector scientist?


is not employed by a university or college (what I think of as academia)
employed within executive arm of government, often a department e.g. Dept of Environment, Dept of Health, Dept of Agriculture
spends subtantial amount of job conducting research
generally not rewarded by academic measures of success and so in some sense decoupled from academic career structures (not saying this is good or bad) 


What do I mean by monitoring?


Keeping statistics and recording trends in number employed, discipline, funding, salaries, career paths, impact on public good or economy etc (while not perfect, there seems to be many reports and organisations tasked with tracking these kinds of things in academia and private sector R&amp;D)


What do I mean by support?


The kind of work done by national academies of science (e.g. AAAS in the US, the Australian Academy of Science, the Royal Society in the UK) and disciplinary societies, professional bodies and the like. Note that their work may or may not be relevant to government scientists, but it is generally not targeted at them. Can anyone share an example of work by a body like this that is specifically targeted to government scientists?
Initiatives, policies, resources aimed at supporting career development and tackling issues. For example just about every university/college and many departments and institutes within them will have their own set of policies, initiatives and resources targeted to early career researchers, whether it's developing writing and publication skills, grant application training, outreach skills, building a career plan, finding a mentor etc.
An issue is that university and college resources, and the staff within them, are often active in the public domain. There are probably a lot of resources out there but government scientists may not be encouraged/authorised to have a public presence and their departments may not prioritise sharing what may be seen as internal resources with the general public.
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).