Let me quote the relevant section from the GPL v3 license, emphasis mine:
2. Basic Permissions.
All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program. The output from running a covered work is covered by this License only if the output, given its content, constitutes a covered work. This License acknowledges your rights of fair use or other equivalent, as provided by copyright law.
You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force. You may convey covered works to others for the sole purpose of having them make modifications exclusively for you, or provide you with facilities for running those works, provided that you comply with the terms of this License in conveying all material for which you do not control copyright. Those thus making or running the covered works for you must do so exclusively on your behalf, under your direction and control, on terms that prohibit them from making any copies of your copyrighted material outside their relationship with you.
The wording of the BSD license makes this implicit:
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
So there's nothing from the GPL and BSD licenses preventing you from running the programs in any scenario, and the GPL even reaffirms so.
Also, note that both GPL and BSD are OSI-approved licenses; that means that the license complies with the Open Source Definition, which states:
6. No Discrimination Against Fields of Endeavor
The license must not restrict anyone from making use of the program in a specific field of endeavor. For example, it may not restrict the program from being used in a business, or from being used for genetic research.
Therefore, this assures that GPL- and BSD-licensed software can be used in any field / kind of work, including governmental.
A different question is whether your particular workplace/scenario/organization allows you to run GPL/BSD licensed software.
And, of course, if you were distributing the programs or allowing other people to use it, the situation would be different. Since you're talking about desktop productivity/graphics software (GIMP, inkscape, etc) this is not the case.