Redirect user to intended URL after login on private site

You can use the wp_login_form() function like this

<?php
    args = 'redirect' => site_url( $_SERVER['REQUEST_URI'] ); //sends them back to the page that the form was on
    wp_login_form($args);
?>

Thanks, where would I put this piece of code?

This code would generate a login form wherever you put it. So for your private page I would use the is_user_logged_in() function like this:

<?php
if ( is_user_logged_in() ) {
    //show the page content you want them to see
} else {
    args = 'redirect' => site_url( $_SERVER['REQUEST_URI'] ); //sends them back to the page that the form was on
    wp_login_form($args);
}
?>