What should I do if emails are not being sent out via Amasty SMTP extension?

As first troubleshooting tip, please check what Amasty SMTP Debug Log and Amasty SMTP Sent Email Logs say. 
Please enable the logs in the main module configuration page.

Where can I see these logs in Magento 1?

• To check the Magento email log, please go to System > Tools and enable Smtp Emails Debug Log.
• Please go to System > Tools and enable Smtp Emails Sent Log.

Where can I see these logs in Magento 2?

• To check the Magento 2 email log, please go to Reports > SMTP  and enable SMTP Debug Log.
• Please go to Reports > SMTP  and enable SMTP Emails Sent Log.