When you work on files saved on external file storage devices such as a USB flash drive, the files are saved to the flash drive unless you save or copy it to the PC yourself. To verify where you are saving the file, use the File/Save as… menu to see where the open file is being saved.
When you connect your flash drive to the PC, there is no automated copying of the content of the flash drive to the PC drive, unless you have installed some kind of file synchronizing application. If you really want to also have these files on the flash drive then you have to save a copy to your PC.
Even for files saved to the flash drive, there might be some traces of the file on PC as you work, however. Let’s say for example you are using Microsoft Word with a file saved on the flash drive. Word keeps a temporary emergency backup copy on your PC as you work, so that if Word or the PC stops unexpectedly it can recover most of your work. When you finish and close the file, some trace of the temporary file might be hidden somewhere on the PC disk, but it would not be like a regular copy of the file someone could find easily or that would show up in a search.2