It's "on" enough to detect the activity needed to wake it back up. But will a powered-off phone still be pinging cell towers or making WiFi requests?

If you cover your phone with an antielectrostatic bag it can't communicate; that is a Faraday cage.

Since people around you will think you are also wearing a tinfoil hard, you had better stick to the phones with hardware switches as sibling comment mentions

Use one of the few phones with hardware kill switches or removable batteries.