The question of population size in the ancient world is one where historians have to work with evidence that is simultaneously abundant and deeply unreliable. Ancient census data exists for Rome and Egypt; estimates for other regions rely heavily on indirect evidence.
The controversy over the meaning of Roman census figures is one of the most technical debates in ancient history. The Republican census counted adult male citizens; the figures range from about 270,000 in the 2nd century BC to over 4 million in Augustus' reign. The 'low count' interpretation holds that the Augustan figure still counts only adult males; the 'high count' interpretation argues it counts all citizens including women and children, implying a massive mid-Republican population collapse that the 'low count' theory explains differently.
Eitan Morley and the Princeton Economic History of the Ancient World series has done the most to systematize what we can and cannot say about ancient population. The broad conclusions — that the Mediterranean world in the early empire had a population of 50-60 million, roughly comparable to the Islamic Golden Age — are reasonably secure. The details are much less so.
The demographic consequences of ancient warfare and epidemic disease are particularly hard to assess. Thucydides' plague killed something like 25% of Athenians. The Antonine Plague may have killed 10% of the Roman Empire's population. The Plague of Cyprian in the 250s AD may have killed even more. These events have no modern parallel in scale, and their economic and institutional consequences are still being worked out.