The mechanisms that produce and power relativistic jets are fundamental open questions in black hole (BH) astrophysics. In order to constrain these mechanisms, we analyze the energy efficiency of jet production η based on archival Chandra observations of 27 nearby, low-luminosity active galactic nuclei. We obtain η as the ratio of the jet power, inferred from the energetics of jet powered X-ray emitting cavities, to the BH mass accretion rateṀ BH . The standard assumption in estimatingṀ BH is that all the gas from the Bondi radius r B makes it down to the BH. It is now clear, however, that only a small fraction of the gas reaches the hole. To account for this effect, we use the standard disk mass-loss scaling,Ṁ (r) ∝ (r/r B ) sṀBondi . This leads to much lower values ofṀ BH and higher values of η than in previous studies. If hot accretion flows are characterized by 0.5 s 0.6 -on the lower end of recent theoretical and observational studies -then dynamically-important magnetic fields near rapidly spinning BHs are necessary to account for the high η ≈ 100 − 300 per cent in the sample. Moreover, values of s > 0.6 are essentially ruled out, or there would be insufficient energy to power the jets. We discuss the implications of our results for the distribution of massive BH spins and the possible impact of a significant extra cold gas supply on our estimates.