The central engine in long gamma-ray bursts (GRBs) is thought to be a compact object produced by the core collapse of massive stars, but its exact nature (black hole or millisecond magnetar) is still debatable. Although the central engine of GRB collapsars is hidden to direct observation, its properties may be imprinted on the accompanying electromagnetic signals. We aim to decipher the generic properties of central engines that are consistent with prompt observations of long GRBs detected by the Burst Alert Telescope (BAT) on board the Neil Gehrels Swift Observatory. Adopting a generic model for the central engine, in which the engine power and activity time-scale are independent of each other, we perform Monte Carlo simulations of long GRBs produced by jets that successfully breakout from the star. Our simulations consider the dependence of the jet breakout time-scale on the engine luminosity and the effects of the detector's flux threshold. The two-dimensional (2D) distribution of simulated detectable bursts in the gamma-ray luminosity versus gamma-ray duration plane is consistent with the observed one for a range of parameter values describing the central engine. The intrinsic 2D distribution of simulated collapsar GRBs peaks at lower gamma-ray luminosities and longer durations than the observed one, a prediction that can be tested in the future with more sensitive detectors. Black hole accretors, whose power and activity time are set by the large-scale magnetic flux through the progenitor star and stellar structure, respectively, are compatible with the properties of the central engine inferred by our model.