I assume the other site is also wordpress?
You could, of course, manually add the stuff with an ad rotator widget, but if you wanted to have it be a little more dynamic, it could be done using some web service calls between the two sites. I don't think there's anything out there like that which is specific to woocommerce.
what info do you want displayed on that site? all or just some of the product data?
I don't think there's anything out there to do this, but I don't think it would be too difficult for me create something like this.
I've never used etsy. Would you want all of your items to be displayed, or just some of them? In the sidebar like an ad, or full pages?