Q. Number of employees in the public or governmental sector by country and in the USA?

ILOSTAT

Page down to "Browse by Subject": Employment. Select Employees: By Sex and Institutional Sector. Click Customize and in the Institutional Sector tab, select only Public.

While the European Union does not provide harmonized statistics, Civil servants in the EU member states provides useful information and links to individual nations.  For OECD nations, Government at a Glance is useful for context.

Government Employment & Payroll from the United States Census Bureau gives more detailed information for the United States.

Sage Data  has statistics for federal (number, average salary, average length of service) and state (number, payroll and part-time hours) government employees.

The Worldwide Bureaucracy Indicators (WWBI) database (2000+) is a cross-national dataset on public sector employment and wages that examines the personnel dimensions of state capability, the footprint of the public sector within the overall labor market, and the fiscal implications of the public sector wage bill. The dataset is derived from administrative data and household surveys, thereby complementing existing, expert perception-based approaches. Encompasses 3 categories of variables: the characteristics of public-sector employment, wages and compensations of these, and the overall wage bill. Together, these provide an important, albeit narrow, picture of the skills and incentives of bureaucrats. Indicators on public employment track key demographic characteristics including the size of the public sector workforce (in absolute and relative numbers), their age, and distributions across genders and academic qualifications. Variables on compensation capture both the competitiveness of public sector wages (compared to the private sector) as well as wage differentials across occupation, gender, education, and income quintiles within the public and private sectors. The indicators on the size of the wage bill offer a glimpse into the structure and affordability of the public sector within the larger economy.