To keep things simple, let's talk about two-qubit states.
A single qubit could have an orthonormal basis ${|0\rangle, |1\rangle}$. But it could also have a different orthonormal basis ${|+\rangle,|-\rangle}$, where $$|+\rangle = \large(\normalsize|0\rangle \small+\normalsize |1\rangle\large)\normalsize / \sqrt{2}$$ $$|-\rangle = \large(\normalsize|0\rangle \small-\normalsize |1\rangle\large)\normalsize / \sqrt{2}$$
No suppose the two states you're trying to distinguish are: $$|\psi\rangle = \large(\normalsize|0\rangle|0\rangle \small+\normalsize |1\rangle|+\rangle\large)\normalsize / \sqrt{2}$$ and $$|\phi\rangle = \large(\normalsize|0\rangle|1\rangle \small+\normalsize |1\rangle|-\rangle\large)\normalsize / \sqrt{2}$$
Here the first qubit in each term is <PERSON>'s and the second qubit in each term is <PERSON>'s.
Let's say <PERSON> does a measurement in the ${|0\rangle, |1\rangle}$ basis. After the measurement the state will be projected into a new state, but which new state depends on the result of <PERSON>'s measurement.
If <PERSON> observes state $|0\rangle$: $$|\psi\rangle \rightarrow |0\rangle|0\rangle$$ $$|\phi\rangle \rightarrow |0\rangle|1\rangle$$
If <PERSON> observes state $|1\rangle$:$$|\psi\rangle \rightarrow |1\rangle|+\rangle$$ $$|\phi\rangle \rightarrow |1\rangle|-\rangle$$
If <PERSON> knows <PERSON> observes $|0\rangle$, then he can distinguish the two states by making a measurement in the ${|0\rangle,|1\rangle}$ basis. 66
Likewise, if <PERSON> knows <PERSON> observes $|1\rangle$, then he can distinguish the two states by making a measurement in the ${|+\rangle,|-\rangle}$ basis.
But if <PERSON> doesn't know the result of <PERSON>'s measurement until after he makes his measurement, then he doesn't know which measurement to make.
If the true state is $|1\rangle|+\rangle$ but Bob guesses it's either $|0\rangle|0\rangle$ or $|0\rangle|1\rangle$, then he's going to make a measurement in the ${|0\rangle,|1\rangle}$ basis. This means he's equally likely to observe $|0\rangle$ or $|1\rangle$, since the true state of his qubit, $|+\rangle$, is an equal superposition of $|0\rangle$ and $|1\rangle$. Note that even if the true state of his qubit were $|-\rangle$ (the only other possibility after <PERSON>'s measurement), <PERSON> would still have been equally likely to observe $|0\rangle$ or $|1\rangle$.
So after comparing results, <PERSON> and <PERSON> realize that <PERSON>'s measurement tells them nothing, and moreover the result of <PERSON>'s measurement by itself does nothing to distinguish the two initial states.
Edit:
I'm going to try to give a more general rule for which states <PERSON> and <PERSON> can distinguish with local measurements only (no communication until afterwards.)
If <PERSON> and <PERSON> aren't going to communicate until their measurements are done, then the combined measurement is effectively chosen before hand. As <PERSON> points out in his answer, this measurement will be a projection into the eigenbasis of $A \otimes B$, where $A$ is a local observable that <PERSON> can measure and $B$ is a local observable that <PERSON> can measure. 66