If $\pi$ is normal, then in fact not only does $\pi$ have secret messages, but it will contain in its digits every possible finite message infinitely often! This is simply because it is part of the meaning of normal number that every finite sequence appears with the expected density. And so in particular, every normal number will contain long blocks of digits that exactly express the collected works of William Shakespeare, and also versions with the plays translated (and mis-translated) into every other language, in all possible ways, and so on.
Indeed, any normal number will contain within it all the theorems of mathematics, with fully correct proofs (as well as with false proofs, in all possible ways).
But of course, this is why you asked about coding on an infinite set, asking for a kind of infinitary regularity property. Leaving aside the particular question about primes, let me consider your latter question: is there a limit to the amount of information that we can extract from $\pi$?
For this, the answer is yes, there is a limit, if we make the problem precise with a particular meaning of what it should mean to extract information. The reason is that $\pi$ is a computable number; there is a computable procedure that will tell us the $n$-th digit. Thus, any procedure that extracts information from $\pi$ by means of a computable procedure inspecting the digits will necessarily be altogether computable. So for example, there can be no computable procedure that extracts information from $\pi$ so as to answer yes-or-no the question of whether a given Turing machine program halts, since the halting problem is not decidable.
More generally, there are only countably many sets that are computable from any given real, no matter the complexity of that real (since there are only countably many programs), and so there is a robust sense in which most sets of natural numbers are not reducible to $\pi$ or any other fixed real in this way.
Lastly, apart from $\pi$, it seems that there are normal numbers $z$ that have the property that the $p(n)$-th digit of $z$ is $0$, where $p(n)$ is the $n$-th prime. Since the primes have asymptotic density $0$, then unless I am mistaken, it appears that we could simply place $0$'s in the prime digits, and choose the rest of the digits randomly, to arrive at a normal number exhibiting your property.