I cannot log in to this forum when the browser is in “private” mode. I am using Safari on macOS 10.13:
File -> New Private Window
Click on the “Log In” button
Enter User and Password, click “Log In”
There is no error message (such as a wrong password), but I am not logged-in. The forum's start page is presented again with “Sign Up" and “Log In” buttons.
You're likely better off reporting this to bugs.webkit.org or Apple's bugreport.apple.com, as there may not be much the forum can do to affect Safari's private browsing behavior.
You are right. I have now verified that it works correctly with Firefox on Windows, so it is a problem of Safari, not of this site. I apologize for the unnecessary noise.