Has Joomla global configuration been properly set to sent email - try a test email from there first.
If the Global email settings in Joomla are correct, perhaps the email is simply going to a spam folder, or through some other spam filtering.
An alternative possiblity, your web host may not allow a "sender" email address that is outside the domain of the actual website or external to their systems e.g. "[email protected]" may not work where "[email protected]" will. But that might be specific to your web host. Usually if you have typical hosting, you can create email accounts in cPanel.