  Anomalous correlators of vector and axial currents which enter the Kubo
formulae for the chiral magnetic and the chiral separation conductivities are
explicitly calculated for free overlap fermions on the lattice. The results are
confronted with continuum calculations in the finite-temperature
regularization, and a subtle point of such regularization for chiral magnetic
conductivity related to the correct counting of the chiral states is
highlighted. In agreement with some previous claims in the literature, we find
that in a lattice regularization which respects gauge invariance, the chiral
magnetic conductivity vanishes. We point out that the relation of anomalous
transport coefficients to axial anomaly is nontrivial due to the
non-commutativity of their infrared limit and the Taylor expansion in baryon or
chiral chemical potential. In particular, we argue that the vector and axial
Ward identities fix the asymptotic behavior of anomalous current-current
correlators in the limit of large momenta. Basing on the work of Knecht et al.
on the perturbative non-renormalization of the transverse part of the
correlator of two vector and one axial currents, we demonstrate that the
relation of the anomalous vector-vector correlator to axial anomaly holds
perturbatively in massless QCD but might be subject to non-perturbative
corrections. Finally, we identify kinematical regimes in which the anomalous
transport coefficients can be extracted from lattice measurements.
