There's not a host on the planet that doesn't provide outbound mail, and why wouldn't you want to send emails? How else would your customers know their tickets had been responded to, random guessing?
More info on ticket IDs: http://osticket.com/forums/showthread.php?t=195(http://osticket.com/forums/showthread.php?t=195)
Do you have your email address set up in osTicket and have the proper outbound address selected in admin panel > settings? If SMTP isn't working, try 'use PHP's built-in mail function' and see if that works. Check your PHP error logs, and the osTicket system logs.
Also make sure that email autoresponders aren't disabled at the department level, which overrides the system default.