  The nature of dark matter (DM) particles and the mechanism that provides
their measured relic abundance are currently unknown. In this paper we
investigate inert scalar and vector like fermion doublet DM candidates with a
charge asymmetry in the dark sector, which is generated by the same mechanism
that provides the baryon asymmetry, namely baryogenesis-via-leptogenesis
induced by decays of scalar triplets. At the same time the model gives rise to
neutrino masses in the ballpark of oscillation experiments via type II seesaw.
We discuss possible sources of depletion of asymmetry in the DM and visible
sectors and solve the relevant Boltzmann equations for quasi-equilibrium decay
of triplet scalars. A Monte-Carlo-Markov-Chain analysis is performed for the
whole parameter space. The survival of the asymmetry in the dark sector leads
to inelastic scattering off nuclei. We then apply bayesian statistic to infer
the model parameters favoured by the current experimental data, in particular
the DAMA annual modulation and Xenon100 exclusion limit. The latter strongly
disfavours asymmetric scalar doublet DM of mass $\mathcal{O}(\TeV)$ as required
by DM-$\bar{\rm DM}$ oscillations, while an asymmetric vector like fermion
doublet DM with mass around 100 GeV is a good candidate for DAMA annual
modulation yet satisfying the constraints from Xenon100 data.
