When using SMTP to send mails (email client or via a script), mails are normally not seen in the Sent folder on the email provider's server. But it does get reflected in the Sent folders of Gmail (Google Apps).
Isn't it mandatory to save the mail in the Sent folder of the server too ? I consider Sent mails as my log for mails sent automatically.
Your mail provider has the choice of multiple ways to provide you access to email. Two of the more common ones are pop3 and imap. It sounds to me like you have pop3, which just pushes emails to the client and forgets about them.
You could have folders for your email, including sent messages, if your provider provides you with an imap server. Otherwise, your option is to change mail providers.
