Just to be clear - When each person makes a measurement, they get only a single value for spin (up or down). If Alice measures the spins of particles on the X axis, she gets up or down for each particle, and about a 50/50 ratio if she measures many. Similarly, if Bob measures spin on the Z axis, he gets up or down for each particle, and about 50/50 for many. If they measure the same axis, X for example, they still get only one spin value for each particle, and about a 50/50 ratio for many. But comparing notes afterward, they will find that their measurements for each particle show opposite spin values.
It should be noted that the 50/50 ratio is approximate - it is a merely a statistical ratio, it is not required that the spin values balance. The entanglement effect merely requires that a given pair of particles have opposite spin values, not that a collection of such particles be exactly half and half.