The livestock sector is the largest source of anthropogenic methane emissions, and is projected to increase in the future with increased demand for livestock products. Here, we compare livestock methane emissions and emission intensities, defined by the amount of methane emitted per unit of animal proteins, estimated by different methodologies, and identify mitigation potentials in different regions of the world based on possible future projections. We show that emission intensity decreased for most livestock categories globally during 2000-2018, due to an increasing protein-production efficiency, and the IPCC Tier 2 method should be used for capturing the temporal changes in the emission intensities. We further show that efforts on the demand-side to promote balanced, healthy and environmentally-sustainable diets in most countries will not be sufficient to mitigate livestock methane emissions without parallel efforts to improve production efficiency. The latter efforts have much greater mitigating effects than demand-side efforts, and hence should be prioritized in a few developing countries that contribute most of the mitigation potential.