I am able to get out to another machine on the network without any issues doing a quick test with 10.2.2 - Are you using UNC paths or a mapped drive letter?
Mine worked both ways - where are you getting an issue? Is the mapped drive letter not showing up under your folder connections or are you receiving an error when executing the extract data?
Is this for all data? or specific datasets?
Can you also copy this data using other data transfer methods?
Is the target file geodatabase new? or is it an existing filegdb you are pointing to?
If it is an existing one - can you try either recreating this filegdb? or instead having the extraction create a new one?
My ordinary procedure is to create the filegdb in the open geodatabase dialog in the wizard, and that doesn't work. I got the same result if I create the filegdb in the catalog window before I open the wizard. Now I tried to create the filegdb in ArcGIS 10.1 Build 3035 (the same version where it works to extract data) and then extract data from ArcGIS 10.2.0.3348. This works! What's the difference beteween a filgdb created in ArcGIS 10.1 and a filegdb created in ArcGIS 10.2?