Thanks, but I hope you are a bit wrong. The "lots of social shares" generally indicate to me IMHO that the site is not high quality, and the image thing just bugs me. I hate images for no good reason.
Interesting idea, but sounds too expensive. I do believe the the image filename and alt/title attributes come into play, though those are easily spoofed so can't be taken too seriously by Google.
For example: Surrounding text, links to the page or image, topic analysis, color analysis, filename, alt attribute, OCR, advanced image recognition, Google Image Labeler (manual labeling).
Reverse image search is probably too expensive to do this for every image found on the web, but Google could surely do this for a subset. Then they could match alt attributes: If two sites use the same image, and one has an alt attribute of "automobile" and the other "car in street" you can ask Bayes how likely it is that these two sites both made fake alt text that does not describe the image.
Basically if Google Images can show you images relevant to your search term, then Google Search can find out if an image is relevant to a topic in much the same way.
Yes, it can improve your rankings, both directly and indirectly.
Directly: It adds rich content to your posts (video's, audio, images). It adds keywords in the image alt attribute and image description. Good images will increase your ranking on Google Images.
Indirectly: If the images contribute to the topic/post (not merely decorative), then you get more user satisfaction, which will increase repeat visits, word-of-mouth and links.
Do the all-things-equal test: Two pages on the same topic (let's say 'Bonobo Monkey'), but one has a picture of the topic. Which one would you find more useful? Two pages of equal quality, yet one has a picture of the author with rich markup next to it, which one would you find more credible?
As with all things, don't overdo it, or use it to manipulate: Adding images when they don't make sense for the topic, copying all your images from elsewhere, spamming the alt attribute with keywords, put a page full with unrelated images etc. Basically do not treat it as an SEO technique, treat it as enhancing your content with media to increase overall quality and user satisfaction.