How can I track download statistics for private files?
Give these modules a try
Download Count

The download_count module tracks downloads of private files attached
to nodes via the core upload or contributed filefield modules for d6
and private core file fields for d7.

File download count

An API for tracking file downloads. Recommended use is by using
file_entity's file/[file:fid]/download callback page. Basically the
difference between this and other file download tracking modules is
this module does not intend to track every single time the file is
viewed. It only wants to track when a file is actually triggered for
download. It is up to the site to implement links to the file download
URL.

AJAX Download Counter

This module uses AJAX to count file downloads. It's optimized for
medium to large sites (such as my own site, OpenGameArt.org) where the
total number of file downloads might be in the hundreds of thousands
or even millions. Using this module has some advantages and some
limitations versus other methods of counting downloads: