The Kindle Cloud Reader, Amazon's browser-based reading app, is now compatible with most (all?) local Kindle stores. I have reports coming in from the UK, Germany, India, and Australia, and a brief check of Amazon's websites for Spain, France, Italy, and Canada confirms that this reading app now works with most local Kindle Stores.
The Kindle Cloud Reader can be found at read.amazon.com, or a local variation thereof.
This app was initially launched in August 2011 as a response to Apple demanding a 30% vig of ebook sales. At that time the app only worked with the main Kindle Store at Amazon.com, and only on certain web browsers (in particular, Safari), but as time went by Amazon expanded support and released a Chrome web app.
That web app has been updated this week, so it too should work with local Kindle Stores. (I'm told Amazon also took the Kindle Cloud Reader and then turned it into the Kindle app for Windows 8, but I cannot confirm that.)
Kindle Cloud Reader has proven to be a fairly useful backup to Amazon's other Kindle apps, but it has never gotten the same attention as Amazon's other apps. It still lacks support for personal docs (I just checked), and it doesn't support all types of content sold in the Kindle Store. And unlike Kobo's competing app, the Kindle Cloud Reader frankly isn't as good as, for example, the Kindle app for the iPad.
Given the lack of attention from Amazon, it's not clear what purpose the Kindle Cloud Reader serves, other than perhaps an early test bed for the idea of a cloud reading app. But at least with the international launch this week readers now have another option.