The radius distribution of small, close-in exoplanets has recently been shown to be bimodal. The photoevaporation model predicted this bimodality. In the photoevaporation scenario, some planets are completely stripped of their primordial H/He atmospheres, whereas others retain them. Comparisons between the photoevaporation model and observed planetary populations have the power to unveil details of the planet population inaccessible by standard observations, such as the core mass distribution and core composition. In this work, we present a hierarchical inference analysis on the distribution of close-in exoplanets using forward models of photoevaporation evolution. We use this model to constrain the planetary distributions for core composition, core mass, and initial atmospheric mass fraction. We find that the core-mass distribution is peaked, with a peak-mass of ∼4M⊕. The bulk core-composition is consistent with a rock/iron mixture that is ice-poor and ‘Earth-like’; the spread in core-composition is found to be narrow ($\lesssim 16{{\ \rm per\ cent}}$ variation in iron-mass fraction at the 2σ level) and consistent with zero. This result favours core formation in a water/ice poor environment. We find the majority of planets accreted a H/He envelope with a typical mass fraction of $\sim 4{{\ \rm per\ cent}}$; only a small fraction did not accrete large amounts of H/He and were ‘born-rocky’. We find four times as many super-Earths were formed through photoevaporation, as formed without a large H/He atmosphere. Finally, we find core-accretion theory overpredicts the amount of H/He cores would have accreted by a factor of ∼5, pointing to additional mass-loss mechanisms (e.g. ‘boil-off’) or modifications to core-accretion theory.