I use a user's browser cache to cache images that are hosted on my server using a .htaccess file and mod.expires.
Here is what I use:
ExpiresByType image/gif A604800
How can I use something similar to cache images from a different domain that I don't have ownership of?
I'm trying to minimize the re-downloading of images a user has already viewed.
No, you can't. To the best of my knowledge, the browser doesn't have any mechanism for fiddling with caching settings.