[EDIT]
0) In the EPR paradox (in fact the CHSH version), based on the hypothesis of local realism , the apparent impossibility of measuring $x$,$z$ polarizations for an entangled state, means that the choice would be between :
a)some interaction exists between the particles, even though they were separated
b) (realism) the information about the outcome of all possible measurements was already present in both particles (hidden parameters).
The first possibility seems incompatible with locality, so Einstein chooses the second possibility (realism), and says that quantum mechanic was incomplete (because we have to add these hidden parameters to the description of each particle).
But experience has showed that Quantum Mechanics violate Bell's inequalities (local realism). So we have to choose between get rid of locality, or get rid of realism. The correct choice is get rid of realism, that is, you cannot consider the 2 particles individually, you have to consider the entangled particles as a whole, you cannot consider the 2 particles separately . This does not mean that Quantum Mechanics violates locality, Quantum Mechanics respects locality. For instance, with entangled particles, it is not possible to send information instantly. Simply, the quantum correlations are stronger than classical correlations.
1) So one first fundamental idea is that the entangled 2-quit state is a whole, and cannot be divided into more little units.
You cannot separate the 2 qbits and the operators acting on them.
2) Alice and Bob can freely choose the orientation of their measurement apparatus, and they always obtain an outcome (your video is wrong about this) which maybe 0 or 1.
3) The fact that quantum mechanics does not respect realism can be seen in the mathematical formalism. For instance, measurements are operators like 2*2 Pauli Matrices. A state is a 2-dimensional complex vector. So applying a measurement to a state, is the same thing that looking at the result of a matrix applying to a vector, for instance :
$$\sigma_x |0\rangle = |1\rangle$$
You see that the measurement change the state, the state after the measurement is not the same that the state before the measurement, so there is no more realism
4) In the case of 2 entangled qbits quantum mechanics, you have to use the formalism of tensorial products. If Alice choose a z-axis measurement, and Bob a x-axis measurement, that means that the measurement operator is :
$$\sigma_z \otimes \sigma_x$$
where $\sigma_z$, $\sigma_x$, are
The above expression is a tensorial product of operators, here it is the tensorial product of the matrices $\sigma_z$ and $\sigma_x$.
It works like this, suppose a separable state $|a\rangle \otimes ~|b\rangle$, then :
$$(\sigma_z \otimes \sigma_x) (|a\rangle \otimes ~|b\rangle) = (\sigma_z|a\rangle) \otimes ~ (\sigma_x|b\rangle)$$
For instance, with your entangled state $|\Psi\rangle= \frac{1}{\sqrt{2}} (|0\rangle \otimes |0\rangle + |1\rangle \otimes |1\rangle )$, it gives :
$$(\sigma_z \otimes \sigma_x)|\Psi\rangle = \frac{1}{\sqrt{2}}(\sigma_z \otimes \sigma_x) (|0\rangle \otimes |0\rangle + |1\rangle \otimes |1\rangle ))$$
$$(\sigma_z \otimes \sigma_x)|\Psi\rangle = \frac{1}{\sqrt{2}}((\sigma_z|0\rangle \otimes ~ (\sigma_x|0\rangle\rangle) + (\sigma_z|1\rangle) \otimes ~ (\sigma_x|1\rangle))$$
$$(\sigma_z \otimes \sigma_x)|\Psi\rangle = \frac{1}{\sqrt{2}}((|0\rangle \otimes ~ (|1\rangle) - (|1\rangle) \otimes ~ (|0\rangle))$$
You see again that the measurement changes the state, so there is no more realism here too.
The assumption (if you cry it or not) "but the particle does have a definite spin, we just don't KNOW what it is, until it is measured! Duh!" is called realism, or in mathier speak, a theory of hidden variables.
Bell's inequalities now say that no theory that fulfills local realism (equivalently that has local hidden variables) can ever predict the correct results of a quantum mechanical experiment.
So we are faced with a problem: Do we give up locality or realism?
Most people choose realism, since giving up locality would totally destroy our conceptions of causality. It is possible that there is a non-local theory that assigns a definite value to every property at all times, but due to its non-locality, it would be even more unintuitive than "particles do not have definite properties".
There is no intuitive explanation for the non-realism of reality (there has to be a way to phrase that better...) because our intuitions have been forged in the macroscopic world which is, to good approximation, classical. But the non-realism is an effect that has no classical analogon, so we cannot understand it in pretty simple pictures or beautiful just-so stories.
Sometimes, we just have to take the world the way it is. (I have assumed that you do not want the whole QM story of non-commuting observables and eigenbases and so on to explain why we, formally from QM principles, expect realism to be false. If I have erred in that respect, just tell me)
Best Answer
In general, for entangled spin pairs magnetic fields are used.
But it's important to note that the states are not the same, as you have stated, but are actually anticorrelated.
So for two entangled electrons, measuring spin along a particular axis, one of them may have a spin "up" and its entangled partner will have a spin "down", the anticorrelated value. Magnetic fields can allow us to measure this.
The Stern-Gerlach$^1$ setup to measure how particles behave in a magnetic field is summarized here. Electrons (and other particles with spin) have a magnetic dipole moment and how they interact with magnetic fields tells us about their spin orientation. Imagine the electron as being a tiny magnetic dipole or "bar magnet", and then imagine what happens if this little magnet is moving in a larger surrounding magnetic field, and how the forces on this magnet would operate depending on the orientation of this magnet (or its direction of spin). Think about one of the electrons moving through one setup and the other through a similar setup in the opposite direction a certain distance away.
Even though both setups may have a spacelike separation, both measurements will come up anticorrelated. And if one is measured before the other, the same will apply even though both electrons are inititally in a superposition of both the up and down state prior to measurement. The initial conclusion was that measuring one state instantaneously "causes" the entangled partner to then be in the anticorelated state, though care must be taken since correlation and causation are not the same thing.
$^1$Note the part that talks about single electrons since the original experiment used silver atoms with an unpaired outer electron, and the results described the spin of this electron.