If I set up a file to be shared and send the link to another person for them to download it, is there any way I can check to see if they did downlad the file?
Unfortunately, this is not a feature of Creative Cloud at this point. However, email notifications are a common feature request in this forum. Feel free to add your own Idea post to promote this feature request.
Here's a feature request for email notifications, specifically for comments on files: http://forums.adobe.com/message/4396814