The He ii transverse proximity effect-enhanced He ii Ly 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 et al., we present a detailed model of the He ii 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 He ii ultraviolet background model, with the added effect of the radiation from a single bright foreground quasar. We vary the age tage and obscured sky fractions obsc of the foreground quasar, and explore the resulting effect on the He ii transverse proximity effect signal. Fluctuations in intergalactic medium density and the ultraviolet background, as well as the unknown orientation of the foreground quasar, result in a large variance of the He ii Ly transmission along the background sightline. We develop a fully Bayesian statistical formalism to compare far-ultraviolet He ii Ly transmission spectra of the background quasars to our models, and extract joint constraints on tage and obsc for the six Schmidt et al. foreground quasars with the highest implied He ii photoionization rates. Our analysis suggests a bimodal distribution of quasar emission properties, whereby one foreground quasar, associated with a strong He ii transmission spike, is relatively old (22 Myr) and unobscured ( obsc < 35 % ), whereas three others are either younger than 10 Myr or highly obscured ( obsc > 70 % ).