Definitely your server because then you can rely upon them. If Flickr or photobucket were to change their URLs, remove your image, or even delete your account, you'd be out of luck. Photobucket also compresses images, resulting in quality degradation oftentimes (or at least, they did when I last used the site in 2009). I don't believe flickr does because they're geared toward photographers.
If it were something like video, which could potentially eat a lot of bandwidth or storage, I could understand using an external solution, but for images, you can't beat storing them on your own server.