We use Herschel 70 to 160um images to study the size of the far-infrared emitting region in about 400 local galaxies and quasar (QSO) hosts. The sample includes normal `main-sequence star-forming galaxies, as well as infrared luminous galaxies and Palomar-Green QSOs, with different levels and structures of star formation. Assuming Gaussian spatial distribution of the far-infrared (FIR) emission, the excellent stability of the Herschel point spread function (PSF) enables us to measure sizes well below the PSF width, by subtracting widths in quadrature. We derive scalings of FIR size and surface brightness of local galaxies with FIR luminosity, with distance from the star-forming main-sequence, and with FIR color. Luminosities L_FIR ~ 10^11 L_Sun can be reached with a variety of structures spanning 2 dex in size. Ultraluminous L_FIR >~ 10^12 L_Sun galaxies far above the main-sequence inevitably have small Re_70um ~ 0.5 kpc FIR emitting regions with large surface brightness, and can be close to optically thick in the FIR on average over these regions. Compared to these local relations, first ALMA sizes for the dust emission regions in high redshift galaxies, measured at somewhat longer rest wavelengths, suggest larger sizes at the same IR luminosity. We report a remarkably tight relation with 0.15 dex scatter between FIR surface brightness and the ratio of [CII] 158um emission and FIR emission -- the so-called [CII]-deficit is more tightly linked to surface brightness than to FIR luminosity or FIR color. Among 33 z <~ 0.1 PG QSOs with typical L_FIR/L_Bol,AGN ~ 0.1, 19 have a measured 70um half light radius, with median Re_70um = 1.1kpc. This is consistent with the FIR size for galaxies with similar L_FIR but lacking a QSO, in accordance with a scenario where the rest FIR emission of these types of QSOs is, in most cases, due to host star formation.
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