It already does, provided you don't have open borders with them. On the side bar there's a notifications tab - they'll show up in there.

Well, one nice improvement they could do is to add filters for all of the notifications.  I wouldnt mind filtering certain builds off, the UP, or even entire planets.  In the OP's situation, he would therefore only see notifications for EXACTLY what he is looking for.  Without filters, notifications on big games are nearly useless, with all of the clutter.

Speaking of notifications, how about a notification when you have birthing rates enabled, and you hit max pop!?!?!?

I also would like to see a delete all notifications button added, I hate doing this individually each turn.