Please note that I use the yt-dlp version, which is a fork of youtube-dl. However, out of habit, I continue to save and use it as youtube-dl instead of yt-dlp.
To download a private video, you need to be logged in; otherwise, you will encounter the following error:
Even with -u option to pass the email, we are still unable to download the video.
As stated in the warning above, the solution is to use existing cookies from the browser where your YouTube account is already logged in. In my case, I use Chrome:
Now, we can successfully download the private video. For further information, please refer to their documentation.