The masses of supermassive black holes are observed to increase with either the total mass or the mean (random) velocity of the stars in their host galaxies. The origin of these correlations remains elusive due to observational systematics and biases that severely limit our knowledge of the local demography of supermassive black holes. Here, we show that the large-scale spatial distribution of local active galactic nuclei (AGN) can constrain the shape and normalization of the black hole-stellar mass relation, thus bypassing resolution-related observational biases. In turn, our results can set more stringent constraints on the AGN radiative efficiency, ɛ. For currently accepted values of the AGN obscured fractions and bolometric corrections, our estimated local supermassive black hole mass density favours mean ɛ values of ~10-20%, suggesting that the vast majority of supermassive black holes are spinning moderately to rapidly. With large-scale AGN surveys coming online, our methodology will enable even tighter constraints on the fundamental parameters that regulate the growth of supermassive black holes.