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).
Is is not 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.
No it is not mandatory, there's two mail protocols, POP3 (Post Office Protocol version 3) and IMAP (Internet Message Access Protocol), when using an email that talks in POP3 protocol, there is no way of telling POP3 to store the email sent...
The workaround was coded and crafted into an email client such as Outlook, Thunderbird, to name but a few, that are using the POP3 protocol.
However, when an email client talks in IMAP protocol, the email server knows where to store it, consider it in layman terms that the IMAP is an enhanced form of the POP3 protocol.
Thus any action carried out on your email client, such as sending mail, will get stored in the designated folder on your machine by your email client, ALSO, the server will dutifully store it in the folder on the server, as a result of the synchronization between your email client and the server.
Usually, an e-mail client such as Microsoft Outlook will receive messages via IMAP4 and send messages via SMTP.
I am unaware of many mail servers that automatically save mail sent through them via SMTP in the server-based Sent folder besides Gmail/Google Apps.
Typically though, once an IMAP4 connection (as opposed to POP3) to the mail server is created for the receipt of incoming mail, one can set the e-mail client to copy sent messages to an IMAP folder. Alternatively, sometimes an automatic BCC can be set up from the e-mail client.
Copying outbound messages to an IMAP folder would cause the synchronization of the local IMAP folder with the mail server and therefore provide a similar effect to the functionality of Gmail/Google Apps. However, in this case, each message sent via SMTP is transferred to the mail server twice: 1) The sending via SMTP. 2) The synchronization between local folder storage in the client and the corresponding folder on the mail server.
See: http://en.wikipedia.org/wiki/Internet_Message_Access_Protocol#Disadvantages_of_IMAP