I have created a playlist of videos on YouTube that I have not made public.
You're going to need to change the privacy setting of the playlist to "Unlisted". Then you'll be able to share a URL with anyone who you want to be able to access the playlist. The playlist won't show up in search, so only those with the URL will be able to access it. You can change the setting back to "Private" again and then no one will be able to reach it again.