According to this wiki, the save game files are located here,
%USERPROFILE%\Documents\Paradox Interactive\Stellaris\save games\$EMPIRENAME+ID\
It does not seem to indicate any difference for different game modes, or if cloud synch is enabled, however the save game files are zipped.
The savegames are zipped. In order to edit them, you'll need to extract them. There are two text files: gamestate and meta. meta includes the information that is shown on the load game screen. gamestate is the actual save game.