  We study bidirectional teleportation while explicitly taking into account an
environment. This environment initially causes pure dephasing decoherence of
the Bell state which assists teleportation. We find that when teleportation is
performed in one direction it is accompanied by a transfer of correlations into
the post-teleportation state of qubit $C$, which results in decoherence of the
state. In the other direction, if no new decoherence process occurs, we find
that not only the state of the qubit but also its correlations with an
environment are being teleported with unit Fidelity. These processes do not
depend on the measurement outcome during telportation and do not differentiate
between classical and quantum correlations. If, on the other hand, the second
teleportation step is preceded by decoherence of the Bell state then the
situation is much more complicated. Teleportation and transfer of correlations
occur simultaneously, yielding different teleported qubit-environment states
for different measurement outcomes. These states can differ in the degree of
coherence of the teleported qubit, but only for an entangling
Bell-state-environment interaction in the first step of teleportation, can they
have different amounts of qubit-environment entanglement. In the extreme case,
one of the teleported qubit states can be entangled with the environment while
the other is separable.
