Problem:

I can't see any css class (related to private posts) to use and I don't know how to set it up (add a class) conditionally in WordPress Archive Template loop.

Solution:

In WordPress Archive loops, you can get post status value with shortcode [wpv-post-status], and use it in [wpv-conditional] shortcode, display different results like this:

https://toolset.com/forums/topic/private-posts-class-of-content-on-frontend-archive/#post-1955557

Relevant Documentation:

https://toolset.com/documentation/user-guides/views-shortcodes/#wpv-conditional

https://toolset.com/documentation/user-guides/views-shortcodes/#wpv-post-status