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We consider settings providing the existence of stable two-dimensional (2D) dissipative solitons with zero and nonzero vorticity in optical media with the quadratic ($χ^{(2)}$) nonlinearity. To compensate the spatially uniform loss in both the fundamental-frequency (FF) and second-harmonic (SH) components of the system, a strongly localized amplifying region ("hot spot",HS), carrying the linear gain, is included, acting onto either the FF component or SH one. In both cases, the Gaussian radial gain profile supports stable fundamental dissipative solitons pinned to the HS. The structure of existence and stability domains for the 2D solitons is rather complex. They demonstrate noteworthy features, such as bistability and spontaneous symmetry breaking. A ring-shaped gain profile acting onto the FF component supports stable vortex solitons, with the winding number up to 5, and multipoles. Nontrivial transformation of vortex-soliton profiles upon either growth of the gain value or stretching the gain profile is observed.