Sync non-Amazon books across Kindle devices and apps
Yes, as of 2016 that is indeed possible. And that’s how it works:
Start with setting up your kindle email address on Amazon, to enable your Kindle devices/apps to receive ebooks via email.
I keep my ebooks on my Mac organized using Calibre, a software which I would dub ‘the iTunes for ebooks’ (or ‘the better iBooks’). It works with epub, mobi, pdf and other filetypes.
Within Calibre, setup sending emails to your Kindle email address. Here’s some help on this.
Then — within Calibre — simply right-click on the book you would like to send to your Kindle Paperwhite or Kindle app, and click ‘Connect/share’, ‘Email to [email address]’ inside the appearing context menu.
The books should then magically appear on your connected devices, like your Kindle Paperwhite or the Kindle iPhone app.
I usually open the book on my Kindle Paperwhite first, check that it works by flicking through a couple of pages, and then switch my Paperwhite off again. Only then I open my Kindle iPhone app, download the book from the ‘Cloud’ onto my device, open the book and check that it syncs to the page I flicked to on my Paperwhite before. You can also force a sync by hitting ‘Sync’ within the hamburger menu.
To get an overview of what you’ve sent to your devices, you can always check on Amazon’s Manage Your Content and Devices section. However don’t expect the non-Amazon books to show up within ‘Books’. You have to select ‘Docs’ from the dropdown to see the non-Amazon books. It appears that the ‘Books’ section is for books bought from Amazon only.
Overall this worked for 99% of the books I have tried.