I just read with some excitement we may be getting push notification capability for Alexa. I'm sure they'll come up with a way to customize what we want, or don't want to be notified with.
This is a huge step if they do it, and I'm very much looking forward to it.
It would also be great if we could custom phrase what we wanted to hear, but I'd be happy just to get canned phrases, rather than nothing at all.
My old X10 set up notified me with custom phrases, I could even put in wav files as a response. But my X10 system was clunky and the devices didn't work all the time. My Smartthings with mostly Z-Wave is running 100%.
I don't think it would be annoying at all for Alexa to talk whenever something happened. But that's me. I'd want to know anytime if the door or window was opening. Or if I left the garage door up. Or someone was at the front door.