  Non-Gaussianities of dynamical origin are disentangled from primordial ones
using the formalism of large deviation statistics with spherical collapse
dynamics. This is achieved by relying on accurate analytical predictions for
the one-point probability distribution function (PDF) and the two-point
clustering of spherically-averaged cosmic densities (sphere bias). Sphere bias
extends the idea of halo bias to intermediate density environments and voids as
underdense regions. In the presence of primordial non-Gaussianity, sphere bias
displays a strong scale dependence relevant for both high and low density
regions, which is predicted analytically. The statistics of densities in
spheres are built to model primordial non-Gaussianity via an initial skewness
with a scale-dependence that depends on the bispectrum of the underlying model.
The analytical formulas with the measured nonlinear dark matter variance as
input are successfully tested against numerical simulations. For local
non-Gaussianity with a range from $f_{\rm NL}=-100$ to $+100$ they are found to
agree within 2\% or better for densities $\rho\in[0.5,3]$ in spheres of radius
15 Mpc$/h$ down to $z=0.35$. The validity of the large deviation statistics
formalism is thereby established for all observationally relevant local-type
departures from perfectly Gaussian initial conditions. The corresponding
estimators for the amplitude of the nonlinear variance $\sigma_8$ and
primordial skewness $f_{\rm NL}$ are validated using a fiducial joint maximum
likelihood experiment. The influence of observational effects and the prospects
for a future detection of primordial non-Gaussianity from joint one- and
two-point densities-in-spheres statistics are discussed.
