I can't reproduce that problem. If I select an image in Design view and press Ctrl-C to copy it, I can paste it into another document that's at the same level in the site. The image is correctly displayed.
If the new page hasn't been saved before I paste the image, I get a broken image icon, but the image appears as soon as the page is saved.
However, it's important for the source and target pages to be in the same folder or at the same level of the site hierarchy. If they're not, the link will be incorrect.
Sorry for the long delay. I'm really not sure what was causing the trouble, but apparently it's nothing some time and a few restarts couldn't fix....