How I Use It
My third through sixth grade students use Edmodo like any social website, usually to share information about their personal lives, websites, games or apps they find interesting, or to ask questions about school projects. Now that we use Google Apps for Education, Edmodo serves this social extension of the classroom community while Google is reserved for academic collaboration, production and feedback of assignments.
Edmodo has been a wonderful way to monitor and instruct students in how to have appropriate interactions in digital contexts. As a class, we can look at which interactions have been helpful (perhaps posting a video that furthers their study of a subject that was covered in class) or not (like a boring "Who's on right now and what are doing?" exchange).
I like having this separate format that can emphasize the social rather than academic aspects of the community, but before we had Google Apps, it encompassed both quite successfully. Students could post work and ask for feedback from me or their colleagues.
The value of the learning is definitely dependent upon how the teacher monitors and uses the website. I've never used any of the paid apps or assessment tools, so I can't comment on their worth.