Not long ago I built an extranet for a client, which was a secure repository of documents. The client wanted to know which files were being downloaded by which users, but Drupal didn't offer that out-of-the-box. So, I had this very modest module written by a nice fellow in the UK.

download_statistics.tar.gz

Untar, upload and install as usual. Then you'll be able to see something like this: 

Your download method has to be set to private, and it's only for 6.x.

I'm not putting this in contrib at drupal.org because I'm not a developer and can't maintain it. Sharing it here is my small thank-you to the Drupal community.