Gnome shell has the ability to have multiple actions associated with an icon, available via right click. However, the firefox desktop file doesn't have any extra actions defined. It'd be handy to have an action that launched a new private window, as the only way to create a private window now (via the gui) is to open a normal window and use a menu there.

This can be added by:

1) Adding an Actions entry to the main section:

    Actions=new-private-window;

2) Defining a section for that action:

    [Desktop Action new-private-window]
    Name=New Private Window
    Exec=firefox --private-window %u

I'm afraid I don't have the ability to translate the name into all the languages otherwise used in the desktop file.

The above will achieve a result similar to the epiphany desktop file. Another possibility is to mimic the gedit desktop file:

    Actions=new-window;new-private-window;

    [Desktop Action new-window]
    Name=Open a New Window
    Exec=firefox %u

    [Desktop Action new-private-window]
    Name=Open a New Private Window
    Exec=firefox --private-window %u

This groups the two 'new window' actions more closely for some reason (the new-window action appears to override the default "New Window" action).