Make sure you've enabled Auto-response and Alerts & Notices in Admin Panel > Settings tab.
If enabled, then it's probably email delivery issue. osTicket's outgoing emails delivery depends on your server settings and email settings. By default, osTicket uses native PHP mail function to send outgoing emails. Make sure php.ini has the correct mail settings for your platform. Alternatively, you can setup SMTP for default outgoing email or each of the email address setup within osTicket.