There has been a 51% drop in emissions from livestock as shown by the FAO statistics, largely due to the shift to a more specialised agricultural livestock system, with still relevant large margin for improvement.

According to the latest European Environment Agency figures, just under 6% of the EU emissions come from livestock GHG emissions. Total agriculture accounts for 9.58% and non-agricultural sectors account for 90.19%. This includes transport, heating, buildings, etc., it doesn’t include land use and land use change.

Thanks to new precision agriculture and advanced agronomic practices the European livestock sector will continue to be able to provide more sustainable and affordable animal-sourced food products to European consumers. 

And the fundamental role of livestock in the circularity of the agriculture sector should not be forgotten, for instance in manure use (also as biogas digestate) for soil fertility and carbon storage and in the consequential avoided emissions to produce agrochemicals and other products derived from oil (i.e. products alternative to leather). 

Sources:
https://ec.europa.eu/eurostat/statistics-explained/pdfscache/16817.pdf
http://www.fao.org/faostat/en/#compare