User restricted only show posts assigned to current user

You search certainly something like Members plugin.

It allow you create a custom Role and You can make your site and its feed completely private if you want or just one or 2 posts. Its like you want.

Or Manually, you can add new roles:

function add_role() {
    add_role( 'private_user', "private user",array(
        'is_able_to_read_private_page'=> true,
        'read'=> true
    ) );
}
add_action('init','add_role');

And Create a custom page and using something like: (ITS AN EXAMPLE)

 <?php /*Template Name: Private page Template*/
 get_header();
 if(current_user_can( "is_able_to_read_private_page")):
    if( have_posts() ):
       while( have_posts() ):
           the_post();
        endwhile;
    endif;
 else:
     echo "Access Denied";
 endif;
 get_footer();?>