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

ILOSTAT

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.

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