If you're not receiving the double opt-in confirmation email or the final welcome email, first make sure there is no error visible after submitting the form.
Also, check the MailChimp for WordPress > Other page in your WP Admin to see if there is any error message in the debug log.
If there is no error in either of those places it means the data is sent to MailChimp successfully. From there it is out of our hands as it is MailChimp.com sending out the opt-in email and/or the final welcome email. Trying with a "fresh" email address often solves the problem.