I mean, if the transmitter is inside the Faraday cage, then there will be electromagnetic waves next to it, and almost not behind the cage. Believe it or not, I put my mobile phone in a coffee can and closed it with a lid, but it continued to work. Not grounded. I needed this in order to accuse the mobile operator of insufficient signal and compensate for the money. At the same time, in my house, it is enough to go 2 meters behind a concrete wall for the signal to disappear altogether.
Metal objects do not absorb the signal, they "amplify" it. The Faraday cage turns into an oscillatory circuit. Even if part of this circuit is grounded, due to the skin effect and inductance, not all of the signal will be absorbed, some of it will come out.
But concrete is not metal. And it perfectly absorbs electromagnetic waves without grounding. In general, concrete is just one type of dense material. The second example is water. Electromagnetic waves affect the electrons inside the substance. Electromagnetic waves transfer energy and vector to electrons. Conversely, electrons transfer energy and vector to electromagnetic waves. The more electrons with different vectors of motion are obtained and the more energy they have, the more waves are damped. In the case of waves passing through a metal, electrons can line up in chains with the same vector and this absorbs electromagnetic waves much less. Therefore, there is no distinction of what would play the role of a body that absorbing waves.
The main thing is that the transmitter should be inside this body, and the sensitive receiver should be outside. And this receiver must register the signal even after turning off the transmitter, because electromagnetic waves must pass through time This is just a thought, just an idea. To do it, you just need a huge piece of concrete or metal, a radio transmitter and a sensitive radio receiver.