There is no Desktop... it is all desktop, except for the web stuff... unless it is done through desktop... then it is blurry
GeoNet Community Structure
BUT since you are using a Geoprocessing tool, the best spot is Geoprocessing. If it were a python problem with a geoprocessing tool that you created, you could put it in Python and 'share' it to Geoprocessing. etc etc
Now. If you can log into the machine as you... and load the toolbox without problem, then the issue is with his profile... remove it. We will start there