Let's say the Hilbert space of one spin, what Bob can measure, is $\mathcal{H}_s$ (spanned by $|\uparrow\rangle$ and $|\downarrow\rangle$). The Hilbert space of the rest of the world is $\mathcal{H}_w$. The total Hilbert space is $\mathcal{H}_s\otimes \mathcal{H}_w$. A general state in this this system is $|\psi\rangle=k_1|\uparrow\rangle\otimes|a\rangle +k_2|\downarrow\rangle\otimes|b\rangle $. The question boils down to how to correctly apply the Born rule to this state when you can only measure $\mathcal{H}_s$ and can't measure $\mathcal{H}_w$.
So Bob doesn't care about the rest of the world. If he looks in his detector and measures the state $|\uparrow\rangle$, he must now project the wavefunction onto this state. $P_s=|\uparrow\rangle\langle\uparrow|$ is the projection operator we want to use on $\mathcal{H}_s$. Bob can't have any physical interaction with $\mathcal{H}_w$, so we act with the identity operator there. $P_w=I$. Acting on psi:
\begin{align*}
(P_s\otimes P_w)|\psi\rangle&= k_1|\uparrow\rangle\langle\uparrow|\uparrow\rangle\otimes|a\rangle +k_2|\uparrow\rangle\langle\uparrow|\downarrow\rangle\otimes|b\rangle \\
&=k_1|\uparrow\rangle\otimes|a\rangle\\
\end{align*}
Of course this has to be normalized.
So yes, by doing absolutely no physics/observation on $\mathcal{H}_w$, we still manage to pick out state $|a\rangle$ over state $|b\rangle$. We did learn something about Alice, but of course we had a huge amount of information inside the wavefunction $|\psi\rangle$ to begin with. To get consistent physics and to predict the probabilities of making a future measurement (perhaps the future measurement is Alice's reaction when they meet back up and say "our spins are opposite, how weird is that?"), Bob must calculate the unitary time evolution of this new state, $|\uparrow\rangle\otimes|a\rangle$, and apply Born's rule again.
If this were a classical probability distribution, this wouldn't be surprising at all. Imagine I take a pair of shoes and perfectly randomly put them in separate boxes. If Bob knows how shoes work, that there is one left shoe and one right shoe (ie, if he knows the probability distribution), then once he opens the box he knows that Alice has the opposite kind of shoe. It's not surprising that he can tell this, because the information was in the probability distribution (which he knew) all along. (I credit John McGreevy for teaching me about classical shoe physics)
A much more convincing demonstration of quantum weirdness is that of "quantum pseudo-telepathy" (that's what Wikipedia calls it anyways, I've never heard that exact phrase before), demonstrating 'success rates' which would be impossible in classical physics.
Quantum pseudo-telepathy is a phenomenon in quantum game theory
resulting in anomalously high success rates in coordination games
between separated players.
