Suppose you have a property Q which certain objects may or may not have. You've seen many of these objects; you know the prior probability P(Q) that an object has this property.
You have 2 independent measurements of object O, which each assign a probability that Q(O) (O has property Q). Call these two independent probabilities A and B.
What is P(Q(O) | A, B, P(Q))?
To put it another way, expert A has opinion O(A) = A, which asserts P(Q(O)) = A = .7, and expert B says P(Q(O)) = B = .8, and the prior P(Q) = .4, so what is P(Q(O))? The correlation between the opinions of the experts is unknown, but probably small. (They aren't human experts.) I face this problem all the time at work.
You can see that the problem isn't solvable without the prior P(Q), because if the prior P(Q) = .9, then two experts assigning P(Q(O)) < .9 should result in a probability lower than the lowest opinion of those experts. But if P(Q) = .1, then the same estimates by the two experts should result in a probability higher than either of their estimates. But is it solvable or at least well-defined even with the prior?
The experts both know the prior, so if you just had expert A saying P(Q(O)) = .7, the answer must be .7 . Expert B's opinion B must revise the probability upwards if B > P(Q), and downwards if B < P(Q).
When expert A says O(A) = A, she probably means, "If I consider all the n objects I've seen that looked like this one, nA of them had property Q."
One approach is to add up the bits of information each expert gives, with positive bits for indications that Q(O) and negative bits that not(Q(O)).
Just consider the limiting case - both are perfect predictors of Q, with value 1 for Q, and value 0 for not Q. And therefore, perfectly correlated.
Consider small deviations from those perfect predictors. The correlation would still be large. Sometimes more, sometimes less, depending on the details of both predictors. Sometimes they will be more correlated with each other than with Q, sometimes more correlated with Q than each other. The degree of correlation with of A and B with Q will impose limits on the degree of correlation between A and B.
And of course, correlation isn't really the issue here anyway, much more like mutual information, with the same sort of triangle inequality limits to the mutual information.
If someone is feeling energetic and really wants to work this our, I'd recommend looking into triangle inequalities for mutual information measures, and the previously mentioned work by Jaynes on the maximum entropy estimate of a variable from it's known correlation with two other variables, and how that constrains the maximum entropy estimate of the correlation between the other two.