Often Facebook friends will share images I think of as “chain letters”, things like “If you know/love a sister/nurse/veteran and (sappy sentiment here) then share this” - do the people who originate these things somehow profit from them?
No.
Sometimes. The answer is not as simple as clockwork would like you to believe. If you get people to share your post, you get positive metrics on facebook. It all works into their new algorithms. The higher shares and conversations about your page the more likely it people will see your page. This works out for companies who are trying to increase their visibility on facebook through less than legitimate methods. blackhat Social Media Optimization is a thing
like this guy
http://www.blackhatworld.com/blackhat-seo/facebook/462013-share-viral-images-facebook-fanpages.html
The people sharing methods like that on black hat sites often aren’t actually making money from them despite what they claim - they’re hoping to spin it off into a scam they can sell.