The short answer is that Axure is not the right tool for this. You’d want something that can interface with a backend handler or database which can handle I/O between your two browser instances–for instance, getting and sending data. By design, Axure doesn’t support data interfaces like this.
The long answer is that you can fake it. You’ll probably need a browser extension to enable a “full screen on dual monitors” view. The basic approach is to launch your prototype in full screen view with no browser “chrome” and place screenshots of empty browser windows where one each would be shown on each monitor, then place your “real” prototype content within the area of each browser window image. You can set up Inline Frame widgets to load individual pages and have each page use a Master image of the browser screenshot as its background, or set the Page STYLE:FILL property to an image and point to your browser screenshot. Use global variables to get two pages “talk to each other” and share common data.