I want to change audio source on my home cinema when Alexa is speaking so I need to be notified when it starts and stops.
No. Currently there doesn't seem to be a way to do that with the Amazon Echo itself—not software based at least. Of course you could open it up and hook a sensor to the speaker but that seems to be not a good way to do what you're asking. Although it would be really IoT to supervise your IoT equipment with more IoT equipment.
However, if you are not opposed to building your own hardware and run Alexa Voice Services from there you could use the events provided by AVS1. That however wouldn't be notifications about the Amazon Echo but only about a device that can act like an Echo.
1: Link dug up by Aurora