Einselection locality in decoherence theory Consider two polarity-wise entangled photons A and B in an EPR experiment. 
The process of measuring the polarization of photon A by Alice is described by the decoherence of the 2-photons system within Alice's apparatus leading to the einselection of a polarization value; let's call this "process A".
Similarly, the measurement by Bob amounts to the decoherence of the same system within Bob's apparatus; this is "process B". 
Let's assume that the measurements happen in spacelike separated frames. 
If einselection is local, how does decoherence theory explain that processes A and B are correlated ?
 A: When you perform a measurement of some observable, the system couples to the measurement apparatus. This spreads information that formerly was confined in the system to the system plus the measurement apparatus, from whence it spreads into the environment. Some of that information is necessary to bring about interference between different values of unsharp observables, and so this spread of information suppresses interference. The observable measured in this way gives the possible outcomes of that measurement - its eigenvalues are selected: einselection. For both subsystems of an entangled system this process of decoherence takes place locally.
Note that decoherence prevents interference between the different possible measurement outcomes. It does not select one of those outcomes and eliminate the others. Both outcomes take place, but they can't interact with one another although they can still sometimes play a role in explaining experimental results,as they do in experiments involving entanglement. The full description of the system in question, and the measurement apparatus, is still given in terms of a set of Heisenberg picture observables, not a single number representing the measurement outcome. Bell's theorem says that if you can represent the state of a system in terms of stochastic classical variables, then physics must be non-local to match the probabilities predicted by quantum mechanics. But in quantum mechanics, a system is represented by its Heisenberg picture observables, which are not classical stochastic variables.
If you have two spacelike separated systems that are entangled with one another and you measure them, then each system decoheres locally. The correlation is established only after the results are compared. They are established by decoherent systems carrying locally inaccessible quantum information: information that is present in a system but does not affects expectation values of measurements on that system alone. See
http://arxiv.org/abs/quant-ph/9906007
http://arxiv.org/abs/1109.6223.
A: I think your concern might really all be about the meanings of the words. So I'll make a very simplified version of the picture so you can see what local means and doesn't mean.
In a Stern-Gerlach device when you measure $\hat\sigma_z$ the incoming particle's beam has some incoming width transverse to the direction it is going. Then the beam widens and splits and on each branch of the split, the beam the spin has changed to be polarized into an eigenstate of $\hat \sigma_z.$
So now you can look at a system of two particles use the x direction for the width of the beam of one particle and the y direction for the width of the beam of particle two. So it starts out like a square and at each point in the square the wave has a value for this joint spin state, say $$\left[\begin{matrix}1\\0\end{matrix}\right]\otimes\left[\begin{matrix}0\\1\end{matrix}\right]+\left[\begin{matrix}0\\1\end{matrix}\right]\otimes\left[\begin{matrix}1\\0\end{matrix}\right].$$
If we send particle one through a Stern-Gerlach then the square widens splits along a vertical line and the spin state becomes $\left[\begin{matrix}1\\0\end{matrix}\right]\otimes\left[\begin{matrix}0\\1\end{matrix}\right]$ on the left square and becomes $\left[\begin{matrix}0\\1\end{matrix}\right]\otimes\left[\begin{matrix}1\\0\end{matrix}\right]$ on the right square.
If instead we send particle two through a Stern-Gerlach then the square gets taller, splits along a horizontal line and the spin state becomes $\left[\begin{matrix}1\\0\end{matrix}\right]\otimes\left[\begin{matrix}0\\1\end{matrix}\right]$ on the bottom square and becomes $\left[\begin{matrix}0\\1\end{matrix}\right]\otimes\left[\begin{matrix}1\\0\end{matrix}\right]$ on the top square.
Each of those is local in the sense that marginals of the other particle didn't change. If you then do the other measurement afterwards, the the whole square deflects left/right or deflects up/down just like a Stern-Gerlach deflects an eigenstate in a particular direction. And the spin state doesn't change.
When you do both at the same time, the it just heads over the the top-right or the bottom left, again the marginals move the same way.
The whole point is that as the spin state changes, the positions change too. That's where that information is getting spread.
A: The question is ill-posed in that it assumes that decoherence explains away the measurement problem, in a local manner. However decoherence only says that probability amplitude phases spread from a quantum system to its environment, effectively entangling the whole thing. So instead of providing a way to get an eventual quasi-classical system, one could say it does the opposite and leads to an eventual quantum system which now includes the environment, with the twist that it may be considered kind of classical in the sense that interference effects are not seen anymore (at least in the limits of experimental error boxes) because phase information is now all other the place, so that probabilities can be handled without having to bother about quantum correlations. But this does not change in any way what EPR has to say about entanglement and nonlocality.
Now as alanf pointed out in its answer, some authors see decoherence as a process in a larger picture where they describe EPR as being local all the way up from the isolated measures to the manifested correlation. See arxiv.org/abs/quant-ph/9906007. But not everyone agrees; see arxiv.org/abs/quant-ph/0312155. 
