There are several independent third party non-government sources for both that validate the government data.
For example on inflation, Adobe Analytics constructs a digital price index that is similar to the Consumer Price Index. And on employment, ADP has a payroll report that precedes the monthly jobs report. Neither are perfect or as robust as the government data, but they are okay.
Our statistical agencies are some of the best in the world and filled with career civil servants.
The Bureau of Labor Statistics does the CPI for inflation. The Bureau of Economic Analysis does the PCE index for inflation.
For jobs data, the BLS also does the payroll survey and has the Census Bureau collect the household survey.
This stuff is just another flavor of conspiracy theory and every few years it seems to get popular again depending on the economy and our politics.