Is writing POD necessary when one is publishing a module or is it optional?

It is both necessary and optional!

It is perfectly possible to publish a module to CPAN with no POD at all. CPAN allows this.

But if users have no idea how to use a module, they won't use it... If users are not going to use it, why bother publishing it? So, treat POD as mandatory for all modules you are going to publish. Would you use a module if you were given no clue about what it does or how you make it do anything?