I believe all Drupal sites make the assumption that the hosting server's PHP is already set up for sending e-mail, but you can add a module like
SMTP Authentication Support to specify your server settings from within Drupal itself: https://www.drupal.org/project/smtp
That's what I do, anyway. I've also had issues getting vanilla e-mailers to work with our own e-mail server and always end up using Google-to-Gmail's SMTP or Sendgrid or something.
Cheers,
Will Matheson
Prince Rupert Public Library