I just chunk them up and bury them in the ground. Something eats them, usually worms. Any organic material improves the soil in the long run.
I prefer to use good clean produce scraps, but the reality is I don't get that much, so I use what I have, which is often conventionally grown. Your lines of what you consider acceptable may differ from mine.
As for what else can be done with them, I cook with them!
Concept Cooking: Watermelon rind experiments: Spoiler alert: they worked WELL! There are links in the first post to recipes for people who use them, the rest of the post is my experiments, since I don't use recipes.