The answer is closely tied to the author's purpose. Amazon can provide significant exposure. From the question's comments, the purpose may be to track the number of readers.
If the real purpose is simply to track the number of readers, Google Analytics or other tracking systems can give you that information with free software. There are other options (meaning other public websites) available which track downloads for you.
On the other hand, tracking your own downloads on a relatively unknown web site as compared to tracking a LOT of downloads on amazon... if you want more exposure, go to the established venues such as amazon.
Wherever you place the book, do work within that site's paradigm or ecosystem. There are ways to get better exposure on Amazon, for example, with lots of books and online tutorials available. If you're going through another venue, find out how to best work within that venue.
Consider multiple venues, multiple e-book file formats, and so on. This becomes an exercise in online marketing or search engine optimization. If tracking is the important consideration, research the various download-tracking options. Once you understand the tracking options available, that may help direct your decision as to how to go about distributing the book or books.
If you have a specific book, niche, or genre in mind, do some searches for free e-books in that genre. That may help you understand the expectations of the book's potential audience. If they're already out there looking for similar books, that fact may guide you in where and how to make it available.
Another way to look at your question is, "How many readers do I gain by putting it on Amazon?" If the book is free, the author is surely under no obligation to publish it EXCLUSIVELY on Amazon. Use amazon in addition to the other options.