If you use a postfix version newer then 2.10, then you need to add the smtpd_relay_restrictions
option as described here:
# With Postfix 2.10 and later, the mail relay policy is # preferably specified under smtpd_relay_restrictions. /etc/postfix/main.cf: smtpd_relay_restrictions = permit_mynetworks permit_sasl_authenticated reject_unauth_destination # Older configurations combine relay control and spam control under # smtpd_recipient_restrictions. To use this example with Postfix ≥ # 2.10 specify "smtpd_relay_restrictions=". /etc/postfix/main.cf: smtpd_recipient_restrictions = permit_mynetworks permit_sasl_authenticated reject_unauth_destination ...other rules...
After that, any sasl authenticated user should be able to send mails through the server using smtp.