iPhone Privacy issues: Siri response / Alerts

I was in a phone conversation moments ago using a headset on my LAND LINE. My iPhone 6+ (iOS 9.3.1) was within arms reach and was connected via USB to my MacPro for backup. The iPhone had been in sleep mode for almost an hour. Suddenly the iPhone lit up, with a partial transcription of my most recent recent remarks in the conversation -and Siri started speaking and made a COMMENT about my conversation -this also appeared in text on the iPhone. I can only guess that somehow the enabled "Hey Siri" feature had been invoked because of something I said. (I never said "Hey Siri" and I did not touch the iPhone) I then disconnected the iPhone from the MacPro -and the text onscreen disappeared. I can't find that text (I don't even have "dictation" enabled) And the dialogue did not appear as a voice memo.

It was as if Siri suddenly joined my conversation. (I was under the impression that "Hey Siri" would only elicit a response if the the iPhone was in active mode (not sleep)

In a separate incident (last night) I was using the same iPhone in Mirror Mode with my Apple TV (current model). I was viewing a streaming Movie on a TV. Suddenly one my personal alerts flashed on the iPhone -AND TV. It was an insignificant alert, but if it had been a sensitive personal item, and if other viewers had been present, they would have seen my alert. This is a privacy issue that I had not anticipated in conjunction with Apple TV.

These issues are of great concern to me as I assumed there were certain aspects of privacy that the iPhone iOS respected. I can only imagine other scenarios why privacy might be inadvertently compromised due to an oversight in the interface or settings.