If you go to test an email from one of the email Message template editors, do you receive the test? If so, it may help to go to Event Espresso > Messages > Message Settings and set the the option for emails to be sent on the same request.
In most cases, setting up a third party service to deliver email will resolve email delivery issues. The built-in mail servers (especially those that are used for shared hosting accounts) can be unreliable. We recommend using a service like Mandrill, Postmark, or any of the other services listed in our documentation for sending transactional email. More info here: