There's an open feature request for the Embedded Media Field to allow editors to link to a live image on any arbitrary site. Although this is technically possible, and easy to implement in the module's framework, this practice is also known as hot-linking, and I'm not particularly interested in enabling that sort of functionality.
However, the discussion on that queue did lead to another interesting possibility. Using file_get_contents (or maybe more efficient method?), I can 'wget' an image and store it to the server. Then the image can just be served locally. I was already examining that possibility for storing (YouTube, Blip.TV, etc.) video thumbnails as well, so I should be able to sign off on two tasks with this mini-project.
It opens up the possibility of copyright violation, but I leave it to the lawyers to fight over that one. And though there are certainly some cases where a hotlink implementation might be okay, I'm personally not interested in encouraging that.