I know this may be beating a dead-horse, but I'm still puzzled by this topic even after reading so many other related SE questions/answers and online articles. I will be using the Copenhagen interpretation, mostly. Also, I have included a lot of information so that a wise responder may beat out every misconception I have on this topic.
First, preliminary information. The wavefunction is a calculational tool in quantum mechanics. It represents the information about a system known by an observer, and the (unitary) evolution of this information obeys Schrodinger's equation. When an observer makes a measurement on the system, what was previously a probability distribution now becomes a definite quantity, and thus the wavefunction "collapses" into the eigenspace of a particular observable. Because the observer can now say that the system has that certain quantity, they then change their wavefunction in order to reflect that appropriately, but nothing forces them to do so. The wavefunction, collapsed or uncollapsed, will still evolve according to Schrodingers equation and still give probability amplitudes for subsequent measurements.
But is the wavefunction then unique to the observer, or must it be common to every potential observer? I know this sounds like a bunch of unimportant hocus-pocus, but because I don't have a good answer to it I don't feel so confident with quantum mechanics as a whole. Please do not misconstrue my words though - I am not saying I do not think nature is quantum. Of course it is, and experiment proves it.
As a tangible example, consider the standard example of two electrons infinitely separated from each other which are together in the singlet state. Using the standard basis defined by an arbitrary "z" axis,
$$|\psi\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow \downarrow\rangle + |\downarrow\uparrow\rangle\right)$$
Now suppose you make a measurement. One of the states is realized.
$$|\psi\rangle = |\uparrow\downarrow\rangle$$
But another experimenter/observer that has not made such a measurement will still describe the system with the first wavefunction, whereas you will describe the particle using the second wavefunction. What's up with this? This seems to imply that, for the other observer who has not yet measured the electron spins, there is such a thing as counterfactual definiteness (this means the state of a system was well-defined prior to measurement). If they measures the electron spins, the first observer can say "I already knew that the electrons were in that state! You could have just asked me!". However, Bell's theorem says that quantum mechanics cannot have counterfactual definiteness (since it is local)!
I feel like somebody may tell me "It doesn't matter which wavefunction you use. They both contain essentially the same information, except that with the second one you will already know the spin-states of the electrons." But I believe this problem goes deeper. The crux of Bell's original paper was that if we assume nature is local, then quantum mechanics can not counterfactually-definite - i.e. the results of experiments can not be defined before the measurements themselves have been made. If the measurements were counterfactually-definite, then they would give different statistics which would satisfy Bell's inequality.
But as for my two aforementioned observers, the electron spins are counterfactually-definite for the first and are not for the second. Why should nature care who measures first?