So APPARENTLY -- and I only found this out recently, but it seems to have been around a while now -- Dreamwidth got image hosting at some point. There's a like... 500MB?... cap, but it even makes thumbnails and lets you set privacy levels per image. It's not very integrated into things, which I will blame for my not having realized -- I'm not sure if the beta post editor has it more integrated, but more universally, you can get to it from Home -> Create -> Upload Images.

Personally I had been using Imgur before finding out about the "new" image hosting, but you can't control privacy on Imgur -- at least not with a guest upload, anyway -- and it's linked to Reddit with all the attendant Redditness.