Andrew- This is an amazing question. You start with some ideas that are a bit out-of-date then jump to an issue so deep that most practicing physicists aren't even aware of it.
First, on the out-of-date part. We only talk about "wave-particle duality" when we're being historical or just a bit lazy. The quantum state is always represented by a wave, more or less localized. As this wave-like state evolves in time, it leads to a variety of different large-scale outcomes- e.g. a state with parts representing both a live cat and a dead cat, in Schrödinger's famous example. Yet we see only a single version of those large-scale outcomes. If we repeat similar experiments many times, we can measure probabilities of different outcomes, and these can be predicted by seeing how much of the quantum state turned into each outcome. So the question becomes why the amplitude of the part of the quantum state should determine the probability.
At one point it was the custom to say that the quantum state "collapsed" to one particular outcome. Since this collapse is outside the usual rules for how quantum states change in time, it was a new process for which we could simply make up new rules, including the probability rule. There has been no success however, in making a picture of that collapse process consistent with the rules of relativity.
These days physicists often say that no collapse occurs. The state simply "decoheres", following the usual rules, into parts which show no interference with each other. Now, since we have no new process, it seems we should be able to deduce the probability rule from basic quantum mechanics.
Here's where it gets interesting. At one point a distinguished physicist published a "proof" in a leading journal that the usual probability rule had to be true. However, hidden in the "proof" was an un-acknowledged assumption: the very same inter-subjective agreement on probabilities that you asked about! So I'd better not use that proof as an explanation, because would just be a circular argument.
So the answer is: we don't know. (Many physicists would disagree with me about that.)
(published on 01/10/2012)