Wave function collapse is not global, it is fictional. Let's suppose that the state is $\alpha|X=0\rangle+\beta|X=1\rangle$, where $|\alpha|^2+|\beta|^2=1$.

When Alice measures the state, an operation is applied that correlates both Alice and the environment with the value of $X$, like so $|X=j\rangle|0\rangle_A|0\rangle_E|0\rangle_B\to|X=j\rangle|j\rangle_A|j\rangle_E|0\rangle_B$. The environment is just everything around the system other than Alice. The $A,E,B$ subscripts stand for Alice, the environment and Bob respectively.

Bob might get the measurement result directly from the system, or from Alice or the environment. In any case, the end result will be $$\alpha|X=0\rangle|0\rangle_A|0\rangle_E|0\rangle_B+\beta|X=1\rangle|1\rangle_A|1\rangle_E|1\rangle_B$$. After the measurement there are two versions of Bob: one version sees 0, the other sees 1. There is no version of Bob that sees both outcomes or some weird mix of 0 and 1, and there is a large literature that explains why this is the case, for an example see

http://arxiv.org/abs/quant-ph/0703160.

The short version is that only information contained in the eigenvalues of some observable, or a subset of such information, can be copied from one system to another. Bob won't see any other information because none of the rest of the information will be copied to him. This follows from quantum mechanics with no collapse postulate. So the collapse postulate is unnecessary for explaining that result.

Wave function collapse is not global, it is fictional. Let's suppose that the state is $\alpha|X=0\rangle+\beta|X=1\rangle$, where $|\alpha|^2+|\beta|^2=1.$

When Alice measures the state, an operation is applied that correlates both Alice and the environment with the value of $X$, like so $|X=j\rangle|0\rangle_A|0\rangle_E|0\rangle_B\to|X=j\rangle|j\rangle_A|j\rangle_E|0\rangle_B$. The environment is just everything around the system other than Alice. The $A,E,B$ subscripts stand for Alice, the environment and Bob respectively.

Bob might get the measurement result directly from the system, or from Alice or the environment. In any case, the end result will be $$\alpha|X=0\rangle|0\rangle_A|0\rangle_E|0\rangle_B+\beta|X=1\rangle|1\rangle_A|1\rangle_E|1\rangle_B.$$ After the measurement there are two versions of Bob: one version sees 0, the other sees 1. There is no version of Bob that sees both outcomes or some weird mix of 0 and 1, and there is a large literature that explains why this is the case, for an example see

http://arxiv.org/abs/quant-ph/0703160.

The short version is that only information contained in the eigenvalues of some observable, or a subset of such information, can be copied from one system to another. Bob won't see any other information because none of the rest of the information will be copied to him. This follows from quantum mechanics with no collapse postulate. So the collapse postulate is unnecessary for explaining that result.