Interestingly, a new study on this topic was published last week in the journal Parasitology.
After an extensive study of Roman and other ancient latrine sites, University of Cambridge archeologist Piers Mitchell has concluded that the incidence of certain easily transferable intestinal parasites such as roundworm and Entamoeba histolytica, a microbe that causes dysentery in humans, was not lowered by Roman advances in plumbing and water delivery, even as compared with the Iron and Bronze Age before it, and the medieval period that came after it.
So, while your celebratory dictator-free kebab might not be carrying any parasites, persistent infection was definitely a problem that the Romans could not effectively combat.