The simplest way to submit data is to use a form. You can create a form, set action, method attribute, add inputs with data, set it value and then do form.submit(). In order to don't redirect user to your site you should also fill a target attribute with name of a hidden iframe. In that case form will be sumbitted "in it". Page woun't reload, but your page will be loaded into iframe (which is hidden). This method allows you to do POST requests.
If you need only GET request, things are even simplier: create an Image object in JS and set src field with the requested url. I don't know how crossbrowser and stable this solution is, but as far as I remember, it works. In fact, you can create and add to page any other element that loads it's content (style, script (and put some codde in response to be evaluated) ).
About crossdomain stuff:
- in case of GET requests you can send information and receive it.
- in case of POST request there is no (as for me) way to get the response.
I suggest you to read something about DOM manipulations in JavaScript and HTML forms to understand it's mechanics and what to do in order to simulate form submission in JS (basically I described it in first two paragraphs).