Some govt employees have pension plans separate from SS, so they pay into those plans rather than SS. Other than that, they pay the same taxes as everybody else, AFAIK.
For Federal civil service, the old CSRS (civil service retirement system), was replaced in the mid-80's with the present-day FERS (federal employee retirement system). CSRS employees didn't have to pay into Social Security, but they also cannot collect from it until they pay into it (used to be 20 quarters, or 5 years......now it's 40 quarters, or 10 years, minimum).
FERS employees have to pay the same taxes as anyone else. Their retirement systems also isn't as generous as the old CSRS, but they had the opportunity to pay into a 401K style plan, called Thrift Savings Plan, aka TSP (and get matching contributions from Uncle Sam). CSRS employees (of which there aren't that many around anymore) eventually got to contribute to TSP, but do not get matching funds from Uncle.
Oh, and no matter how many studies they make, that allegedly show that federal civil service employees make twice as much as private company employees, try to remember that a large percentage of federal civil service employees live around high cost of living areas, like Washington DC, San Diego, Seattle, etc. Especially around DC, many of them are VERY high ranking executive-rank employees (think Secretary of Defense, Commerce, State, etc, and their direct deputies, down 3 or 4 levels of management). Executive series employees do make pretty good bank, but it's because of the level of responsibility (think CEO, COO, company presidents and vp's, etc). Many more of them are managerial and supervisory GS pay grades, who also make pretty good money, especially as the GS pay scale pays cost of living allowance in high cost of living areas.
Take away all those people, and just count the working men and women, doing their day to day jobs, and you'll find that the vast majority of them could make much better money "on the outside".