Short answer:
Probably zero (statistical mode) for many battles. Higher numbers for encirclements and slaughter of captives. Possibly higher numbers for legionaries than hoplites. Highest numbers are likely among cavalry chasing down routed enemies.
Discussion:
Based on the numbers we have -- which have problems-- many soldiers would likely not have killed anyone, or perhaps one person. One would hypothesize a fairly high variation-- elite troops in the front line and cavalry chasing down broken enemies might have killed many. We have a suggestion of a high casualty rate among centurions due to their exposed positions, they likely killed more than average as well. Other formations toward the rear might have participated little. But the mode, the number of "kills per soldier" most frequently found -- that would likely be zero in many if not most engagements.
Adrian Goldsworthy portrays this plausible picture of battle (note "plausible", not proven or provable):
We may picture a line of men in contact with an enemy unit, with the majority of soldiers fighting very cautiously, gaining the maximum protection from their shields, watching their opponents, and only occasionally delivering a weaker blow, exposing as little of their right arm and side as possible. A minority of men would fight far more aggressively, attacking their opponents with powerful blows from their scutum, and delivering savage thrusts and cuts with their swords
This is consistent with the idea of a fairly high degree of variation in involvement in combat. Some folks uninvolved, others tentative, and others both killing a lot and themselves more likely to be killed. The question of "what do soldiers actually _do_ in battle" is as much a contemporary one as an ancient one. Ancient legionary and hoplites were in formation, with less opportunity, perhaps, to just keep their heads down-- but even in a legion's ranks, there likely were big distinctions in initiative and aggression.
The statistical mean kill per combatant would often have been below one (# of combatants > # of enemies killed) for many engagements, it's only in a few particularly bloody battles where kill counts are reported as higher. There are battles in history where you get combat fatalities in excess of 1x a force's strength, eg an army of 10,000 that kills 20,000 in battle-- not impossible, but not the norm.
There are scholars who have attempted to extract "average casualty data" from the reports we have in ancient sources, with a suggestion that hoplite battles were less bloody that Roman battles:
"Krentz has studied casualty statistics for hoplite engagements, and has concluded that the victors lost an average of 5 per cent and the defeated side an average of 14 percent of their strength. Casualties among the losers in Roman battles were frequently far heavier, with over half the defeated army often being killed or taken prisoner according to our sources"
[Krentz' casualty data is here ]
We have a number of accounts that include slaughter of captives _after_ battles, this could have resulted in much higher numbers for some. In general, though, armies didn't and don't "fight to the last man" -- numbers will vary, but an ordinary unit with %10-20 casualties may well break and run, or be pulled back. Elite units may fight on with higher casualties; poor quality ones may break with less. Krentz notes that lower casualty counts are associated with battles when pursuit of the defeated didn't occur, specifically"Syracuse in 415 and at Coronea in 394"; implies that much of the killing took place in the pursuit of a broken enemy.
In the ancient world one can point to sources that attest particularly sanguinary military defeats-- Cannae in 216 BCE where Hannibal encircled and substantially destroyed a Roman army is a case where we get reported Roman casualties substantially in excess of reported Carthaginian forces, implying an average "kill count" in excess of one for Carthaginian forces. That's all dependent on our sources being accurate in their troop and casualty counts; they might or might not be. We're substantially dependent on Polybius for numbers, if he's wrong, then our calculations will be too.
There are other battles where our sources report an extraordinary slaughter. Sources for Alesia, for example, report that on the order of 70,000 Roman forces and auxiliaries are said to have killed some 250,000 Gauls, and taken another 40-50,000 prisoner. If that indeed happened (it's possibly exaggerated); you'd be looking at Roman soldiers killing, on average 3 to 4 Gauls; some would have done less, some many more. There's reason to be skeptical of such numbers. It's very hard to know what to make of casualty counts that seem extraordinary-- at Gaugamela, for example, we've got sources telling us that Alexander's army lost somewhere between 100 and 1000 men, and that they killed some 40 to 90 thousand enemy. Reading Arrian, many of these casualties were fleeing the battlefield-- whether you call that a death in combat or something else is your choice of terms.
Alesia and Cannae illustrate one of the characteristics of a battle that might result in higher casualties -- encirclement, where an enemy couldn't retreat. Similarly the capture of a city, with ensuing slaughter-- very difficult to know what are the deaths of civilians or captives vs combatants.
Against the example of these bloodbaths, we have examples of generals who skillfully achieve their goals without a pitched battle, perhaps none so famous as Quintus Fabius Maximus Verrucosus who gains the sobriquet "Cunctator" (= "the delayer"), for his strategy of skirmish and delay, an operational plan which frustrates Hannibal, but produces no victory. To have fought under Fabius' command -- you might well have marched and countermarched for weeks and months and neither caused nor suffered much bloodshed.
Sources:
Daly, Gregory. "Cannae: The Experience of Battle in the Second Punic War" [Routledge:2002]
Goldsworthy, Adrian "Cannae" [Cassell:2002]
Goldsworthy, The Roman Army at War, 100 BC-AD 200 (1996),
Sabin, Philip. “The Face of Roman Battle.” The Journal of Roman Studies, vol. 90, 2000, pp. 1–17.
P. Krentz, 'Casualties in hoplite battles', GRBS 26/I (I985), 13-20.
also -- on the tendencies of soldiers to avoid participation (at least in modern warfare) see an old thread here
with thoughtful comments by u/Badgerfest and a now deleted user.