  We study interacting 1D two-component mixtures of cold atoms in a random
potential, and extend the results reported earlier [{\it Phys. Rev. Lett.} {\bf
105}, 115301 (2010)]. We construct the phase diagram of a disordered Bose-Fermi
mixture as a function of the strength of the Bose-Bose and Bose-Fermi
interactions, and the ratio of the bosonic sound velocity and the Fermi
velocity. Performing renormalization group and variational calculations, three
phases are identified: (i) a fully delocalized two-component Luttinger liquid
with superfluid bosons and fermions (ii) a fully localized phase with both
components pinned by disorder, and (iii) an intermediate phase where fermions
are localized but bosons are superfluid. Within the variational approach, each
phase corresponds to a different level of replica symmetry breaking. In the
fully localized phase we find that the bosonic and fermionic localization
lengths can largely differ. We also compute the momentum distribution as well
as the structure factor of the atoms (both experimentally accessible), and
discuss how the three phases can be experimentally distinguished.
