Hello.
I suppose the easiest way is to add a custom code into 
\CMSWebParts\Newsletters\NewsletterSubscriptionWebPart.ascx.cs, 
SaveSubscriber() method. You can write/call a code to send e-mail after the subscriber is saved:
// Save subscriber info (line 755 in 5.5R2)
SubscriberProvider.SetSubscriber(sb); 
// here you can send e-mail via api.
You can send e-mail message using following example code:
CMS.EmailEngine.EmailMessage msg = new CMS.EmailEngine.EmailMessage();
    msg.From = "mymail@domain.com"; // use valid e-mail
    msg.Recipients = "mymail@domain.com"; // use valid e-mail
    msg.Subject = "Custom BizFrom e-mail";
    msg.Body = "The value of the FirstName field: "
                  + CMS.GlobalHelper.ValidationHelper.GetString(
                      this.viewBiz.BasicForm.DataRow["FirstName"], "N/A");
    CMS.EmailEngine.EmailSender.SendEmail(msg);
Best Regards,
Radek Macalik