I thought I'd pop in and mention this set of lectures (now a book, huh) from Scott Aaronson on quantum mechanics and quantum computing: http://www.scottaaronson.com/democritus/ Particularly Lecture 9 goes into the basic mathematics of quantum mechanics, and explains why it's more appropriate to consider it an extension to probability theory using complex numbers and call them amplitudes, rather than classic probabilities. Not being a physicist myself I don't know if this is really helpful in clearing up professional scientists' intuitions, but it's helpful for me at least in grasping the basics.

You're on the right track by noting that quantum theorists can use probability to communicate their findings. A single classical probability like 0.7 is a summary, not the whole picture. For the whole picture, you need a log of observations with time stamps, which is a tremendous amount of data because taken to the extreme it's a balancing act between maximizing the data you know about you and your surroundings and minimizing the number of planck unit time steps to collect the data. Even with more realistic amounts of data it's still a lot to pack around, so you can summarize it by constructing a probability distribution which is more convenient mathematically, and you can summarize that probability distribution into a handful of numbers if you need to because those are even easier to pack around. (Like the single probability 0.7, or two numbers u and s that characterize a Normal distribution, etc.)

So if you think of probability as just summarizing your past observations and constraining your expectations for future observations (because what else can?), I think it's easy to see how two people can have different probabilities that represent a belief in a proposition or an abstract system or something else. If both are being honest (I'm packing a lot of assumptions in that word since this is basically Aumann's agreement theorem) and the probabilities are different, there are a couple reasons why: either at least one of them has information (observations) the other one does not, or they started out with different prior probabilities. In the former case, they can reach agreement by sharing data. In the latter case, they can eventually reach agreement by obtaining more data or choosing a common prior. With enough successive Bayesian updates, you can arrive at the same conclusion regardless of where you started. (In practice, of course, things are more hairy and not this simplistic, which is what the submitted pdf is addressing in part.)

I find it hard to grok what it means for something to have a physical frequency property. I can understand it as a statistic of an abstract system, i.e. a probability, but in light of Many Worlds, I don't think a frequency can fundamentally be a property of something in the same way energy can be. But since the energy of a photon is related to the photon's "frequency" via its "wavelength", is energy really fundamental or is the relation just a convenient means of making the wave mechanics useful? I've read Feynman tell how the whole idea of forces is just a somewhat circular simplification that's not really fundamental, even though sometimes it seems like it is. While I have a high degree of certainty about the physics accurately describing reality as she is, my meta-belief / confidence about that high certainty isn't nearly as high simply because I feel a lot of confusion about the ultimate nature of reality. But at least at a higher level than fundamental physics, it's clear to me that saying a "fair coin" has p=0.5 per unique side is really just summarizing we don't have enough knowledge of the flipper and the environment it is flipped in to predict the result with certainty in the way we can predict the result of the output of a logic OR gate with low- and high- voltage inputs with certainty. This is different than the uncertainty of which branch you are within a Many Worlds universe, where it's more of a physical resolution problem rather than one of needing more knowledge, similar to not being able to find out what happens between planck unit time steps (if anything happens and if the question means anything).

Ben Goertzel's book Probabilistic Logic Networks is another resource I'd recommend. Sorry if my branching reply is a bit big, but I'm constantly trying to clarify things for myself as well. ;)