  We propose two models for the creation of stable dissipative solitons in
optical media with the $\chi^{(2)}$ (quadratic) nonlinearity. To compensate
spatially uniform loss in both the fundamental-frequency (FF) and
second-harmonic (SH) components of the system, a strongly localized "hot spot",
carrying the linear gain, is added, acting either on the FF component, or on
the SH one. In both systems, we use numerical methods to find families of
dissipative $\chi^{(2)}$ solitons pinned to the "hot spot". The shape of the
existence and stability domains may be rather complex. An existence boundary
for the solitons, which corresponds to the guided mode in the linearized
version of the systems, is obtained in an analytical form. The solitons
demonstrate noteworthy features, such as spontaneous symmetry breaking (of
spatially symmetric solitons) and bistability.
