  JWST has revealed a population of compact and extremely red galaxies at
$z>4$, which likely host active galactic nuclei (AGN). We present a sample of
434 ``little red dots'' (LRDs), selected from the 0.54 deg$^2$ COSMOS-Web
survey. We fit galaxy and AGN SED models to derive redshifts and physical
properties; the sample spans $z\sim5$-$9$ after removing brown dwarf
contaminants. We consider two extreme physical scenarios: either LRDs are all
AGN, and their continuum emission is dominated by the accretion disk, or they
are all compact star-forming galaxies, and their continuum is dominated by
stars. If LRDs are AGN-dominated, our sample exhibits bolometric luminosities
$\sim10^{45-47}$ erg\,s$^{-1}$, spanning the gap between JWST AGN in the
literature and bright, rare quasars. We derive a bolometric luminosity function
(LF) $\sim100$ times the (UV-selected) quasar LF, implying a non-evolving black
hole accretion density of $\sim10^{-4}$ M$_\odot$ yr$^{-1}$ Mpc$^{-3}$ from
$z\sim2$-$9$. By contrast, if LRDs are dominated by star formation, we derive
stellar masses $\sim10^{8.5-10}\,M_\odot$. MIRI/F770W is key to deriving
accurate stellar masses; without it, we derive a mass function inconsistent
with $\Lambda$CDM. The median stellar mass profile is broadly consistent with
the maximal stellar mass surface densities seen in the nearby universe, though
the most massive $\sim50$\% of objects exceed this limit, requiring substantial
AGN contribution to the continuum. Nevertheless, stacking all available X-ray,
mid-IR, far-IR/sub-mm, and radio data yields non-detections. Whether dominated
by dusty AGN, compact star-formation, or both, the high masses/luminosities and
remarkable abundance of LRDs implies a dominant mode of early galaxy/SMBH
growth.
