As discussed in [Lo and Popescu 1999], it is always possible to distill entanglement from pure bipartite states using a one-way protocol. The gist of the idea, as I understand it, is Proposition 1 in the paper, which states that for any bipartite pure $|\Psi\rangle$ and PVM $\{P_\ell^{\rm Alice}\}_\ell$, we have $$(I\otimes P_\ell^{\rm Alice})|\Psi\rangle = (U_\ell^{\rm Alice}\otimes U_\ell^{\rm Bob})(P_\ell^{\rm Bob}\otimes I)|\Psi\rangle$$ for some unitaries $U_\ell^{\rm Alice}$, $U_\ell^{\rm Bob}$ and PVM $\{P_\ell^{\rm Bob}\}_\ell$. This means that at whatever step of the protocol, we can avoid Alice measuring her state and telling Bob the result, by directly having Bob perform some other measurement, followed by a unitary $U_\ell^{\rm Bob}$ on his side, followed by him telling Alice to perform some unitary $U_\ell^{\rm Alice}$ on her side. Iterating this reasoning, we can avoid having Alice do anything altogether except applying some local unitary at the end of the protocol.
On the other hand, [Bennet et al. 1995] previously (among many other things) gave a general protocol to perform entanglement distillation of pure states, which I also discussed in How are the singlets distilled in Bennett et al.'s 1995 protocol?. The idea is that if Alice and Bob share $n$ copies of $|\psi\rangle=\alpha|00\rangle+\beta|11\rangle$, then if they both measure in a PVM that only distinguishes between states with different numbers of excitations (eg $\{\mathbb{P}_{00},\mathbb{P}_{11},\mathbb{P}_{01}+\mathbb{P}_{10}\}$ for $n=2$ qubits), their measurements are maximally correlated, and conditionally to each outcome they know they share a maximally entangled state of some Schmidt rank (which depends on the measurement outcome at each run). As I understand it, this is a "zero-way communication strategy", as Alice and Bob don't need to communicate at all to make it work.
Finally, in a different paper [Bennet et al. 1996] argue that for mixed states, one-way communication cannot be sufficient to distil pure state entanglement. To this end, they consider (see Section IV) the Werner state $W_{5/8}\equiv \frac12|\Psi_-\rangle\!\langle\Psi_-|+\frac12 (I/4)$. Locally, the authors argue, each copy of $W_{5/8}^{\otimes n}$ just looks like a maximally mixed state to Alice, and therefore "she cannot distil even one good EPR pair from an arbitrarily large supply of $W_{5/8}$ states. On the other hand, they argue that using two-way strategies some nonzero amount of entanglement can be distilled.
The paper contains a number of strategies that they argue can be used to distil entanglement from states like $W_{5/8}$, but I can't quite extract what's the idea behind them. Is there some general intuition or reasoning that allows to understand why two-way communication should allow to extract entanglement from something like $W_{5/8}$?
