A large cage probably doesn’t need to be grounded to prevent a relatively weak signal from escaping, as attenuation would be high due to the amount of material involved. Smaller cages may radiate the signal after some attenuation.

Edit: reading some more about it, cages that are close to the radiating element may experience capacitive coupling, and this is what can cause an ungrounded cage to serve as an antenna. A larger cage, with the radiating element farther away from the cage, is less likely to experience this. In either case grounding should reduce this risk.

Well, what does it mean to be "grounded". There isn't something special about the voltage potential of Earth.

If a Faraday cage blocks interstellar signals only if one part of it is stuck in a ball of mud and rock... well, I have some questions.

There is the possibility of the ground being a return path to the transmitter, but if that were effective, radio infrastructure would interfere world-wide, and you could transmit through the earth's core. And even that argument would suggest that the Faraday cage should be floating, not grounded.