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Bell's theorem states that any physical theory that incorporates local realism and the no-conspiracy assumption cannot reproduce all the predictions of quantum mechanical theory.

Hence, we cannot develop a theory of local hidden variables if we assume non-conspiracy.

If we assume the freedom of choice of the experimenter (to, for example, measure the x-axis rather than the y-axis in an experiment), how do we maintain determinism in, for example, the MWI? Doesn't the very fact that she has freedom to choose necessitate that her actions are not physically determined?

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The assumption of "freedom" in the Bell inequality derivation is a somewhat misleading statement. It is not an assumption of freedom in the philosophic sense. Rather, it is the statement that the experimenter's choice of measurement basis should not be correlated with the quantum state they are measuring.

For example, one could set the measurement apparatus such that for each run of the experiment the measurement axis is determined by the last digit of the Dow Jones at that moment. Then the relevant assumption of the Bell inequality derivation is that the laws of nature not somehow determine the stock market so that it fluctuates in sync with the secret changes in the true orientation of the entangled spins. You can see why theories that break this assumption are often described as "conspiratorial." You can also see that science itself depends on making assumptions like this. You can't trust the result of any experiment if you start to suppose that, say, the laws of nature are set up so that when you do a particular experiment you always suffer a hallucination that makes you think that you got the opposite result. This is why most people tend to dismiss this possibility. That said, some people do try to construct theories that violate this assumption without being as egregiously conspiratorial as the above example- see, e.g., this recent question.

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