Properly shielded equipment uses different methods to 'break the cage'. It's been many decades, but some of the heavily shielded designs I did in the 80's involved opto-isolators. Yes, that's right. Want to avoid radiating information ? Use light.
Keep in mind that the structure of the faraday cage depends on the frequency of the data being transmitted. It does not have to be unbreakable tin foil. Properly sized metal mesh will also do the job. Just ask anyone who tries to get a Wifi signal through an old wall with expanded metal lath and plaster.