  The scattering of kinks and low-frequency breathers of the nonlinear
sine-Gordon (SG) equation on a spatially localized $\mathcal{PT}$-symmetric
perturbation (defect) with a balanced gain and loss is investigated
numerically. It is demonstrated that if a kink passes the defect, it always
restores its initial momentum and energy and the only effect of the interaction
with the defect is a phase shift of the kink. A kink approaching the defect
from the gain side always passes, while in the opposite case it must have
sufficiently large initial momentum to pass through the defect instead of being
trapped in the loss region. The kink phase shift and critical velocity are
calculated with the use of the collective variable method. Kink-kink
(kink-antikink) collisions at the defect are also briefly considered, showing
how their pairwise repulsive (respectively, attractive) interaction can modify
the collisional outcome of a single kink within the pair with the defect. For
the breather, the result of its interaction with the defect strongly depends on
the breather parameters (velocity, frequency and initial phase) and on the
defect parameters. The breather can gain some energy from the defect and as a
result potentially even split into a kink-antikink pair or it can lose a part
of its energy. Interestingly, the breather translational mode is very weakly
affected by the dissipative perturbation, so that a breather penetrates more
easily through the defect when it comes from the lossy side, than a kink. In
all studied soliton-defect interactions the energy loss to radiation of
small-amplitude extended waves is negligible.
