  The HeII transverse proximity effect - enhanced HeII Ly{\alpha} transmission
in a background sightline caused by the ionizing radiation of a foreground
quasar - offers a unique opportunity to probe the emission properties of
quasars, in particular the emission geometry (obscuration, beaming) and the
quasar lifetime. Building on the foreground quasar survey published in
Schmidt+2017, we present a detailed model of the HeII transverse proximity
effect, specifically designed to include light travel time effects, finite
quasar ages, and quasar obscuration. We post-process outputs from a
cosmological hydrodynamical simulation with a fluctuating HeII UV background
model, plus the added effect of the radiation from a single bright foreground
quasar. We vary the age $t_\mathrm{age}$ and obscured sky fractions
$\Omega_\mathrm{obsc}$ of the foreground quasar, and explore the resulting
effect on the HeII transverse proximity effect signal. Fluctuations in IGM
density and the UV background, as well as the unknown orientation of the
foreground quasar, result in a large variance of the HeII Ly{\alpha}
transmission along the background sightline. We develop a fully Bayesian
statistical formalism to compare far UV HeII Ly{\alpha} transmission spectra of
the background quasars to our models, and extract joint constraints on
$t_\mathrm{age}$ and $\Omega_\mathrm{obsc}$ for the six Schmidt+2017 foreground
quasars with the highest implied HeII photoionization rates. Our analysis
suggests a bimodal distribution of quasar emission properties, whereby one
foreground quasar, associated with a strong HeII transmission spike, is
relatively old $(22\,\mathrm{Myr})$ and unobscured $\Omega_\mathrm{obsc}<35\%$,
whereas three others are either younger than $(10\,\mathrm{Myr})$ or highly
obscured $(\Omega_\mathrm{obsc}>70\%)$.
