Somehow, it doesn’t seem to be possible to plan post for my subscibers (visibility private) as they get published private automatically as soon as I save them.
Is there any way I can keep wordpress from doing so?
Planned private posts get published immediately
The problem is that changing the post visibility to private invokes a status change, in a way private is not only a visibility but also a publishing status. Or to quote the WordPress Codex, section »Content Visibility – Private Content«.
To achieve what you want you probably have to work with the publish_post or even better the publish_future_post hook. Another possibility is to work with Post Status Transitions and the according hooks.
Have you checked your timezone settings under Settings -> General? One of the things that can cause this is if the schedule date is in the past according to the configured timezone but not according to your time.