Do people these days still keep a computer on in their house? Assuming yes..

IPv6 should make it easier for mobile and home devices to connect. Then devices could pair by requiring that the user take a picture of a qr code on the computer screen using the phone. That's all it takes to authenticate the computer and the phone to each other when they connect forever after. Perfect forward secrecy could even easily be used for this connection. We can use all this to guarantee security but unfortunately not anonymity because a local passive attacker would be able to see when the user gets a message: it would be immediately* relayed to the phone.

*One might argue that you could introduce a randomized delay. But it would still be discernible after a number of trials and the number of trials wouldn't need to be very large. The only solution I'm aware of is to send data of definite size to the phone every x minutes thus an attacker wouldn't know what is real data and what is just filler. But I don't like this solution.