The non-communication theorem is an assumption, intended to be self-consistent with the constancy of c assumption. That's all. And around these self-consistent assumptions is built the self-consistent mathematical framework. Do away with the constancy of c assumption, and the non-communication assumption no longer stands.
The second postulate is that the speed of light has the same value c in all inertial frames of reference. It may well turn out that this is a quantum-mechanical phenomenon related to quantum tunneling and analogous to Heisenberg's uncertainty principle. In this context, a particle's motion through space is not the simple, linear vector as commonly understood in Newtonian physics. Michelson-Morley's famous "failed experiment" has done away with the ether, but relativity theory need not be the only alternative explanation. This QM perspective can factor in the role of the observer - nay, it SHOULD factor in the role of the observer - and an observer travelling at the speed of light will intercept photons in exactly the same manner as if he were stationary... no violation of the second postulate, by any observer at any speed. This is an important possibility that might help explain numerous phenomena, such as the quantum eraser experiment.
References to random information, or stochastic correlation, seem generally to be poorly thought through. Think of radio waves and their carrier frequency, to finish up as sound coming out of your radio's loudspeaker. If your "random information" can be made to follow a pattern, then you might still be able to transmit information. Perhaps there is scope for converting random information into some kind of carrier base over which information might be transmitted, much like a radio carrier frequency being used to broadcast sound.