  The counterflow dynamics of two correlated impurities in a double-well
coupled to an one-dimensional bosonic medium is explored. We determine the
ground state phase diagram of the system according to the impurity-medium
entanglement and the impurities two-body correlations. Specifically, bound
impurity structures reminiscent of bipolarons for strong attractive couplings
as well as configurations with two clustered or separated impurities in the
repulsive case are identified. The interval of existence of these phases
depends strongly on the impurity-impurity interactions and external confinement
of the medium. Accordingly the impurities dynamical response, triggered by
suddenly ramping down the central potential barrier, is affected by the
medium's trapping geometry. In particular, for a box-confined medium repulsive
impurity-medium couplings lead, due to attractive induced interactions, to the
localization of the impurities around the trap center. In contrast, for a
harmonically trapped medium the impurities perform a periodic collision and
expansion dynamics further interpreted in terms of a two-body effective model.
Our findings elucidate the correlation aspects of the collisional physics of
impurities which should be accessible in recent cold atom experiments.
