In the Everett Interpretation (often called the Many Worlds Interpretation), when an 'observer' system interacts with an 'observed' system in a superposition of states, the result is that the observer transitions to a superposition of states correlated with those of the observed system.
So you start with two particles in a correlated superposition of up and down states, and two observers in a "not seen anything yet" state.
The first observer observes the first particle, and enters a superposition of "observer A sees up" and "observer A sees down" states, that are correlated with the particle he observed, and hence also with the other particle. The states are orthogonal, and so don't interact with one another, and cannot 'see' one another. It is as if each observer seeing each possible outcome was in his own separate world, although in fact they're all in the same world, in superposition, like the up and down states of the particle are in the same world, in superposition. No extra universes are created. It just looks like that seen from the inside.
The second observer observes the second particle and enters a superposition of "observer B sees down" and "observer B sees up" states, correlated with the particle, and hence the other particle, and hence the other observer.
When the two observers get back together to compare notes, the "observer A sees up" state can only see and interact with the "observer B sees down" state, and the "observer A sees down" state can only see and interact with the "observer B sees up" state. Their observations are correlated, because their wavefunctions are.
The observers end up correlated ('entangled') the same way the particles were. If you need to think of it in terms of 'worlds', then you could say there are just two worlds. Initially, the particles each split, half going into the 'up/down' world and the other half going into the 'down/up' world. Then when the first observer observes the first particle, he joins the particle in splitting between the two worlds. Then the second observer makes his observation and he too splits between the same two worlds. Then when the observers get back together again, each perceives only the version of the other in the same world.
I think the 'many worlds' analogy for it is confusing - there being 'many worlds' for one entity can still be a 'single world' for another. Everett himself talked about the wavefunctions of the different systems becoming correlated, which I think is a much better way of looking at it.
The same phenomenon happens in classical physics. When oscillators are coupled (i.e. allowed to interact by applying forces on one another), they enter what are known as 'normal modes of vibration'. The overall motion is a sum of a number of individual solutions to the joint equation of motion, each of which is orthogonal to all the others, and so vibrates independently as if none of the others existed, and in which the motions of all the oscillators are correlated with one another.
There is a classic demonstration of the effect in which a length of string is tied loosely between two supporting posts, and then two pendulums of nearly equal period are hung from the string. When you push one, it swings freely, but then slows to a halt as all the energy is transferred to the other pendulum, then it in turn slows as all the energy is transferred back. Analysis shows that it is a sum of two normal modes of vibration, one where the two pendula swing together in synchrony, and the other where they swing oppositely to one another. The normal modes each have a slightly different frequency, and the sum shows 'beats' as they move in and out of phase.
The whole point of the Everett Interpretation is that it doesn't introduce any extra metaphysical effects beyond pure unitary quantum mechanics. There are no extra universes being created. There is no spooky faster-than-light action at a distance. There is no random indeterminacy. There is no "wavefunction collapse" triggered by mysterious, vaguely-defined and often disturbingly vitalist causes. The theory is realist, local, deterministic, fully specified and (at least in regard to the microscopic world) already accepted as true by the physics mainstream. It simply points out (or so Everett claims) that there is no 'Measurement Problem' to be solved - unmodified quantum mechanics already predicts that quantum observers will see classical outcomes.
