How to make a custom template in WordPress work as a password protected page?

Question

I'm building a page with a custom template.

The thing is, I need this page to be password protected, or at least accessible to logged in users, but even if I set it as such (Private/Password protected) in the New Pages section in WordPress Administration, it won't display the menu entry nor the content (if Private) or it would show the page contents immediately (if Password protected).

I've read somewhere that the_content() function is what makes this work, but as you can guess, my custom template doesn't use the_content() at all, and it's all based on custom content.

Do you happen to know how can I (re)implement these two options?

Answer 1

Removed the content of my answer. TheDeadMedic's answer is the correct one.

Answer 2

There's a much more elegant and reliable approach to checking if a post is private;

if (post_password_required() {
    // It's protected and they haven't entered a password, so ask for one:
    the_content();

} else {
    // It's not protected or they have entered a password
}