The built in recorder always currently dumps the recording in the user's home directory. This is:

 A) Good because it doesn't show up anywhere that you might be recording
 B) Bad because it's pretty mysterious

What probably would make the best UI sense is to record into a mktemp'ed file in /tmp, then have a dialog when the recording finishes to select where to save it. This probably wouldn't be in ShellRecorder would have a signal to say "I've saved it to <X>", and you connect to that signal from JS to do the GUI.

(Some issues if the recording got really large - /tmp might be on tmpfs so might have size limits, and if it's not the same partition as /home moving it to Desktop or Documents might require copying the bits.)

Simpler would be to just allow configuring the file name that it saves to via environment variable, or command line option to --gnome-shell. All the internals are there to do that, just not the actual configuration. 

Simpler yet would be to move the default location from ~ to ~/Desktop so the user would see where the recording was dropped. The downside is (as mentioned above) if you are taking a series of short moves within a session, you have to move that aside before recording another.

(Related is the question of whether the movie should show up in recent documents or not)