But presumably you could have the Pi listen for a trigger word or whistle or whatever using software running locally, and when triggered, kick over to the Alexa API?
[1] https://ifttt.com/products/do/button [2] http://nodered.org/
Also you need a microphone array to do it reliable (the Echo has 7 microphones).
On the software side, supposedly they're using Nuance for recognition. Nuance isn't cutting edge: In the tests I've done, Nuance has a Word Error Rate (WER) that's 10%-20% higher than Google's, but it's still much better than something like Pocketsphinx or any other open source recognizer.
There are a lot of factors that go into making a speech interface a good experience for users: Good recognition accuracy even with background noise, good voice activity detection (even with background noise), very accurate word spotting, low latency. It's hard to hit all these things well enough to make the interface usable.