Campylobacter jejuni and Campylobacter coli are the biggest causes of bacterial gastroenteritis in the developed world, with human infections typically arising from zoonotic transmission associated with infected meat. Because Campylobacter is not thought to survive well outside the gut, host-associated populations are genetically isolated to varying degrees. Therefore, the likely origin of most strains can be determined by host-associated variation in the genome. This is instructive for characterizing the source of human infection. However, some common strains, notably isolates belonging to the ST-21, ST-45 and ST-828 clonal complexes, appear to have broad host ranges, hindering source attribution. Here whole-genome sequencing has the potential to reveal fine-scale genetic structure associated with host specificity. We found that rates of zoonotic transmission among animal host species in these clonal complexes were so high that the signal of host association is all but obliterated, estimating one zoonotic transmission event every 1.6, 1.8 and 12 years in the ST-21, ST-45 and ST828 complexes, respectively. We attributed 89% of clinical cases to a chicken source, 10% to cattle and 1% to pig. Our results reveal that common strains of C. jejuni and C. coli infectious to humans are adapted to a generalist lifestyle, permitting rapid transmission between different hosts. Furthermore, they show that the weak signal of host association within these complexes presents a challenge for pinpointing the source of clinical infections, underlining the view that whole-genome sequencing, powerful though it is, cannot substitute for intensive sampling of suspected transmission reservoirs.