The duality has something to do with strength of interaction of a system with its environment, which may or may not consist largely of a piece of measurement apparatus of which we are consciously aware. In short, the duality arises from fixating on two extremes of behaviour: strongly coupling with the environment, or not. (Realizing this doesn't necessarily simplify our understanding of QM, but it is the theme underlying the dualities you have noted.) 

What all of the interpretations agree on is this: a system which is isolated evolves according to the Schrödinger equation, and a system which is interacts strongly enough with a macroscopic system that we can observe a difference in the behaviour of that large system does not. These are two polar extremes of behaviour; so it is not in principle surprising that they exhibit somewhat different evolutions. This seems to me where the duality comes from: stressing these two opposite poles. 

 * In the Copenhagen interpretation, the "quantum" systems are the isolated ones; the "classical" systems are the large macroscopic ones whose conditions we can measure. Nothing is said about the regime in between.
 * In von Neumann's description, the evolution of isolated systems is by the Schrödinger equation; ones strongly coupled to macroscopic systems get projected. Again, nothing is said about the regime in between.

"Decoherence" and "Many-Worlds" are not easily distinguishable interpretations of quantum mechanics (indeed, in Many-Worlds, the preferred basis is thought to be selected by decoherence, though this must still be demonstrated as a technical point); but:

 * In Decoherence-in-Many-Worlds, there is a duality in between worlds we experience, and worlds we do not (which is related to the question of what precisely defines a 'world');
 * In Decoherence-without-Many-Worlds, there is a duality between degrees of freedom which are highly entangled with the environment (which appear to behave stochastically), and those which are independent of the environment (which may behave deterministically if you perform the right measurement).

The fuzziness of the boundary between these two situations, in fact, is a symptom of the fact that "not completely isolated" is not the same as "strongly coupled to the environment". There is, presumably a gradient; and furthermore, you get to choose what the boundaries of "the environment" (that part of the world which is just too big and messy for you to try to understand) are. So, if a physical system is only a *little* leaky, or is interfered with *only slightly* by the outside world, you can try to account for this outside meddling, and so describe the system as one which may be *somewhat less* leaky.

Some of the projects of interpretations of quantum mechanics are trying precisely to describe the two extremes, and so everything in between, using a monism of dynamics. Many-worlds, for instance, seems to shrug at the question of why we only perceive one world out of many, but wholeheartedly believes that *all dynamics* is in principle unitary, and is trying to prove it. And Bohmian Mechanics already has monism, albeit at the cost of faster than light signalling between particles by way of the quantum potential field — albeit signalling which manifests macroscopically only as *correlations*, for essentially thermodynamical reasons — which understandably puts most people off.

Note that there are also dualisms in science, historically and in modern times, outside of quantum mechanics:

 * historically: terrestrial and celestial mechanics (subsumed by Newtonian mechanics)
 * historically: organic versus inorganic matter (subsumed once the chemistry of carbon started to become well-understood)
 * currently: gravity (treated geometrically) versus other elementary forces (treated by boson mediation)
 * currently: "hard sciences" (theories of the world largely excluding human behaviour) versus "soft" sciences (theories of the world largely concerning human behaviour)

Any time you have two different models of the world which do not seem obviously compatible, but which do (at least somewhat successfully) describe systems well in some domain, there is a sort of duality between those two models. The dualities in our current understanding of quantum mechanics are somewhat unique in that they concern exactly the same systems, and in the fact that interactions in one of the regimes ("strong coupling with the environment") seems to be the only way for us to obtain information about what happens in the other ("weak coupling with the environment")!