What security issues are you talking about? Universal Clipboard (via Handoff) operates over BLE locally. iCloud is used in part of the process to facilitate key exchange.
If the government/other actor is going through the effort of getting Apple to somehow disclose this key (assuming that it would even be possible) and lurk outside your window, you have bigger things to worry about.
Read the link I provided. Apple documents the security mechanisms of how a lot of their services work. Universal Clipboard (and handoff) does not use iMessage.
I've read a lot of their whitepapery security stuff but my understanding and differentiation of terms like "Continuity", "Handoff", "Universal ______", "Airplay" vs "Screen Mirroring" etc needs revision and reconsolidation.