Is it possible to search for private posts

You could do this by using a custom template called better-search-template.php in your theme’s folder.

And modelled along the lines of: https://github.com/WebberZone/better-search/blob/master/templates/template.php

The plugin fetches posts using Better_Search_Query which extends WP_Query for pulling the search results.

You could pass an additional argument called post_status similar to what you’d do with WP_Query.

An alternative option is to create a function to filter better_search_query_args to set the post_status argument.