"At some point in the future, streaming music and personal assistant options will be added, though it’s not clear whether Amazon customers will have to log in with their own account information"
From Amazon's FAQ:
4. How do I delete Alexa voice recordings?
You can delete specific voice recordings associated with your account by going to History in Settings in the Alexa App, drilling down for a specific entry, and then tapping the delete button. Or, you can delete all voice recordings associated with your account for each of your Alexa-enabled products, by selecting the applicable product at the Manage Your Content and Devices page at www.amazon.com/mycd or contacting customer service.
Deleting voice recordings may degrade your Alexa experience. If you delete voice recordings, we will also remove Home Screen Cards related to those voice recordings. However, removing a Home Screen Card from your Alexa App does not delete voice recordings. If you have changed your default marketplace while using an Alexa-enabled product, you will need to perform this step for each marketplace to delete all voice recordings associated with your account.