https://developer.mozilla.org/en-US/docs/Mozilla/Command_Line_Options?redirectlocale=en-US&redirectslug=Command_Line_Options

Command Line Options - Mozilla

Example Usage: C:\PROGRA~1\MOZILL~1\firefox.exe [ options ... ] [URL]

      where options include:

 -h or –help		Print this message.
 -v or –version	Print Firefox version.
 -P <profile>		Start with <profile>.
 -migration		Start with migration wizard.
 -ProfileManager	Start with ProfileManager.
 -no-remote		Do not accept or send remote commands; implies -new-instance.
 -new-instance	Open new instance, not a new window in running instance.
 -UILocale <locale> Start with <locale> resources as UI Locale.
 -safe-mode		Disables extensions and themes for this session.
 -console		Start Firefox with a debugging console.
 -jsconsole		Open the Error console.
 -browser		Open a browser window.
 -new-window <url> Open <url> in a new window.
 -new-tab <url>	Open <url> in a new tab.
 -preferences	Open Options dialog.
 -search <term>	Search <term> with your default search engine.
 -setDefaultBrowser Set this app as the default browser.

Not too sure what your problem is but it seems to work as expected. You need to place the url after the option. From a CLI

firefox -private http://example.com 

works for me to open the url in a private mode window.

I note from http://manpages.ubuntu.com/manpages/xenial/en/man1/firefox.1.html It explains:

SYNOPSIS
      firefox [OPTIONS] [url]
....
OPTIONS

...

     -private
             Start firefox in private browsing mode