Opposite spins yes, definite no. They are in a superposition of states. In one state, particle A is up while particle B is down, while in the other state, particle A is down and particle B is up. When the pair is created, they are in an entangled state, and only after measurement do they assume definite outcomes.
The crux of the matter is that this particular state is rotationally invariant. So we should get the same results whether we measure spin about the z axis or the x axis or any other. Now, if the two spacelike separated detectors each measure spin along the z axis and repeat the experiment many times, they find perfect correlation, even though their separate readings are random. On the other hand, if they take readings along orthogonal axes, they get zero correlation, while still getting individually random readings. It would seem that the two particles must conspire somehow. How is the measurement axis of one detector communicated to the other particle in order to achieve the correct correlation?
What people seem to miss out on is that somehow the choice of measurement axis needs to be communicated from one detector to the other, if we are to hold onto a classical or local hidden variable description. This, of course, breaks a lot of other things.