We need a giant public "mailstore" where everybody (or at least a large pool of people) put the encrypted messages with encrypted recipient information. Everybody who uses this mailstore gets a copy of every message posted to it, but unless they are the intended recipient with the right private key they can't make heads or tails of it.
Maybe usenet could be drafted into handling it. Usenet already handles terabytes of encrypted data on a regular basis nowadays.
Wait, that's so wrong. Why don't we just fix our laws. ...
Wait, we will always have bad actors or regimes, do we just design for that?