Since the 3.3 Software Update for Kindle, Amazon has enabled support for syncing personal documents through Whispersync. The only thing missing, however, is that personal documents do not sync to Kindle Apps (iOS app, for example).
I've been using a workaround I discovered to send documents to my iPhone so I can read them on that, as well as my Kindle. This does not, however, sync your progress or notes. This method only allows you to make the documents accessable on the other apps. For me, that's good enough, I just have look up the last location and go to it before I start reading.
When I try to deliver a document, the only available option is my Kindle. My iPhone and Cloud Reader show in the list but are disabled. Using Chrome's Developer Tools to inspect the HTML select element (below), I found that it has a unique ID for each device. Now, if I can find out what the ID is for my iPhone, I might be able to change the value of this option in the DOM and it just might send the document to my iPhone!
To get the iPhone's ID, I head over to the Manage Your Devices page and inspect the listing for my iPhone.
Going back to the Personal Documents page and using Developer Tools to change the value of the ID to that of my iPhone's ID, I was able to send personal documents to my iPhone! This does not work for the Cloud Reader.