AJAX post not working with HTTPS
If you are doing the AJAX post from a http page to a https URL then the Cross-Domain policy kicks in because the protocol is also part of the origin specification, as it is described here. The browser will refuse to make the AJAX call, so that's why you're not seeing any traffic.
A solution is discussed here:
Ajax using https on an http page
So your best bet is the Access-Control-Allow-Origin header which should be supported on most modern browsers now.
So make your server add the following header to the responses:
Access-Control-Allow-Origin: https://www.mysite.com
If for some reason you cannot enforce this, then the only choice left would be JSONP.