We address the problem of non-asymmetric Slepian-Wolf (SW) coding of two correlated non-uniform Bernoulli sources. We first show that the problem is not symmetric in the two sources, contrarily to the case of uniform sources, due to the asymmetry induced by two underlying channel models, namely additive and predictive Binary Symmetric Channels (BSC). That asymmetry has to be accounted for during the decoding. In view of that result, we describe the implementation of a joint nonasymmetric decoder of the two sources based on Low-Density Parity-Check (LDPC) codes and Message Passing (MP) decoding. We also give a necessary and sufficient condition for the recovery of the two sources, that imposes a triangular structure of a subpart of the equivalent matrix representation of the code.