  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.
