Choose User to email when adding a new post?

You would want to add a custom meta box that lists all users on the site. Not sure off the top of my head how you would create one that lists users, but it shouldn’t be too hard.

You would then run a hook on wp_insert_post that verifies which users have been checked, and sends each of them an email. Something like:

add_action('wp_insert_post', 'my_function');
function my_function()  {
$user_ids = "whatever you saved it as in your custom meta box"; // grab the user ids of who you want to email in an array 

    foreach($user_ids as $user_id)
        $user_data = get_userdata($user_id);
        wp_mail($user_data->user_email, "your subject", "your message");
    }
    return;
}