Custom Plugin w/ Custom Post Types – Custom Posts Are Showing at all Privilege Levels – Is this possible to adjust?

Yes this is possible. Add the following to either your plugin code or the theme’s functions.php file:

function mypo_parse_query_useronly( $wp_query ) {
    if ( strpos( $_SERVER[ 'REQUEST_URI' ], '/wp-admin/edit.php' ) !== false ) {
        if ( !current_user_can( 'update_core' ) ) {
            global $current_user;
            $wp_query->set( 'author', $current_user->id );
        }
    }
}

add_filter('parse_query', 'mypo_parse_query_useronly' );

Reference original source here

This will limit the posts displayed to the each author’s own posts only.