Amazon says it saves these recordings in the cloud until you ask to delete them. But when I checked, it had stored conversations back to March 2016, but I had an Echo since September 2015, so it’s not perfectly consistent. Either way, you should assume that anything you say to Alexa will be stored by Amazon indefinitely.
But, understand: Both Amazon’s Alexa privacy FAQ and an Amazon spokesperson says Amazon does not record your conversations all of the time, but only when you speak the wake word “Alexa.”
In other words, Amazon promises Alexa isn’t always listening to you.