One of the most common ways to do this is AJAX. Here's how you perform an AJAX post request using jQuery:

<script type=text/javascript>

  $.post('/remote-url', {xml: yourXMLString });

</script>

On the server side you process it like any other POST request. If you're using PHP it's $xml = $_POST['xml'];

The biggest limitation of AJAX is that you're only allowed to make requests to the same domain the document has been loaded from (aka cross-domain policy). There are various ways to overcome this limitation, one of the easiest one is JSONP.

UPD. For cross-domain requests an extremely simple (though not universal) solution would be:

(new Image).src = 'http://example.com/save-xml?xml=' + escape(yourXMLString)

This will issue a GET request (which cannot exceed 2KB in Internet Explorer). If you absolutely need a POST request or support for larger request bodies you can either use an intermediate server-side script on your domain or you can post a dynamically created html form to iframe.