You are asking the right kind of question, and the main thing I want to say is to warn you that a huge amount of stuff has been written on this and only about $0.00001$ percent of all that stuff is worth reading.
These questions go to the heart of what we mean by a 'state' when talking about quantum systems. Most people try to frame their discussion by invoking the notion of a quantum state and writing down state vectors (kets in Dirac notation). This mostly sets one up to fail because one has adopted from the start the very language which leads to confusion when dealing with correlations of the type considered in Bell inequalities.
One can avoid some of the muddles by refusing to do that, and instead adopt an approach with more of a quantum field theory flavour. That is, one makes the calculation like an input--output problem, or a type of scattering process. One does not make statements about collapse of wave function, one simply calculates the quantum amplitude for whatever overall process from start to finish one wishes to know about.
The main thing I would want to say is that in the entangled scenario neither party has an individual state in and of itself, for the physical quantity under discussion (e.g. direction of spin angular momentum). One must train oneself to stop thinking of it as if it did---as if the spin was "partly up and partly down" or something like that, or could "collapse".
In order to unpack that final comment a little further, consider the following sequence of events:
Sequence A:
- entangled pair is created at some source O, and sent to places A and B---one part to A, the other part to B
- an operation is applied at A
- the pair is brought together at C where it is manipulated and measured.
Now consider the following sequence of events:
Sequence B:
- entangled pair is created at some source O, and sent to places A and B---one part to A, the other part to B
- an operation is applied at B
- the pair is brought together at C where it is manipulated and measured.
I have in mind an operation that did something---it was not an identity operation. For example, consider an operation such as a rotation which had the effect that the final state, observed at C, is orthogonal to the state prepared at O.
Now the important thing about an entangled state is that sequence A and sequence B can have the same overall effect. That is, the change caused by the operation in the middle of the sequence is the same change. It causes the same result at C in the two cases. So we have two operations which have one and the same effect. This means that, at least as far as their effect on this entangled pair is concerned, they are one and the same operation, just carried out different ways, like if you turn a coin over using your left hand or your right hand, the effect on the coin is the same. But the thing being compared to a coin here is at two different locations---the spacelike-separated locations A and B.
It is mistake to say that in sequence A the operation adjusted some physical property of the particle at A. It did not, because the same result can be obtained by doing nothing at all at A and instead following sequence B. So one must say that the operation, whether at A or B, adjusted a physical property of the pair, a property which cannot be assigned to either of A or B on their own.
The kind of property we are talking about here could be the direction of the spin angular momentum, and the associated magnetic field in the case of a magnetic dipole, or it could be polarization for a photon or internal energy for an atom. Those are all experimentally realised examples. But in principle it could be anything at all.
Finally, instead of the sending the particles back together at the end as in my two sequences, one could instead have them interact with some other parties such as measuring devices and then bring the results of those interactions together at C. This is the scenario usually described in experiments of this kind. The above, with suitable modifications, still applies.
I will finish by doing something I have deliberately resisted doing. This is to write down an entangled state in Dirac notation. As I already warned, to do that is already to embark on a misleading way of thinking, because it suggests that each part of the system has its own state. But that is the very thing that is not true. The Dirac notation here is
$$
| \psi \rangle = \frac{1}{\sqrt{2}} \left( |\downarrow \rangle_A
|\downarrow \rangle_B + |\uparrow \rangle_A
|\uparrow \rangle_B \right)
$$
or some other entangled state. It gives the impression that the particle at A has a physical property called direction of spin, and so does the particle at B, and the two are correlated. But I claim this is not so. To see why, apply a rotation to the particle at A. The state becomes
$$
| \psi_1 \rangle = \frac{1}{\sqrt{2}} \left( |\uparrow \rangle_A
|\downarrow \rangle_B + |\downarrow \rangle_A
|\uparrow \rangle_B \right).
$$
So the spin at A has changed, right? Well no. If we return to $| \psi \rangle$ and apply a rotation at B then we get
$$
| \psi_2 \rangle = \frac{1}{\sqrt{2}} \left( |\downarrow \rangle_A
|\uparrow \rangle_B + |\uparrow \rangle_A
|\downarrow \rangle_B \right)
$$
and the crucial thing here is that
$$
| \psi_1 \rangle = | \psi_2 \rangle.
$$
It is the attempt to put into words what is going on here that fills so many pages of writing.
The essence of the calculation by John Bell, and which is also on view in the above, is, I think, that we must abandon the way of thinking in which every physical property is physically represented or carried as a property of an individual thing separated off in its own little spacelike region. The physical world simply is not like that. It is like that in most of science, but not in this little corner of quantum physics---a corner that proves to be not so little because it is involved in things like phase transitions and chemical reactions, and possibly in processes significant to biology such as the way enzymes work.
On this view, the result of a process such as a measurement at A is not best described by saying that it immediately caused a change at B. Rather, one would say that the spin direction and associated things such as the local magnetic field is here physically carried by a two-particle entity that is present at A and B and that cannot be correctly understood as two individual things. It can be understood as two things in some of its properties such as mass, but not for entangled properties such as spin direction in this example. In short, one has a single physical embodiment, not two, and other parts of the physical world can interact with this physical embodiment at either A or B or both. The physical property of spin direction is here physically embodied in a non-local way.
