so as you already pointed out ancient (classical) warfare was generally fought by tightly packed formations of phalanxes and later more mobile legions by the romans, but this would probably not have had a big impact on your knowledge of the battle for the simple reason that battle line could span across a large front and thus you could already be in the process of being encircled and not know it, that is probably also why euripides told us of "intestinal problems" of some phalangites.

in we focus closer onto the classical 5th century bc phalanx we would probably come across a generally untrained (that goes for every greek city, including sparta) civilian militia and some pesky mercenaries like scythian archers for athens, and that played a huge role on who would win the battle, or from a hoplites perspective who was winning, and the reason for that is because one man fleeing for whatever reason from the battlefield could have detrimental consequences for the entire formation because it could lead to a mass rout. that is exactly why phalanxes would generally have the more experienced men both in front and in the back, to incite and hold the younger and unexperienced troops from running away.

and that is really the gist of it, if you are advancing, then you are probably winning, if people start running then you probably aren't, if your leadership got killed, then you probably aren't winning although someone on the other side of the battlefield wouldn't have known until later on.

there were obviously some exceptions, most notably sparta that used flutes to give orders, athough i don't think that there is any evidence to suggest that they were actually used during the battle itself. yeah i just rechecked to be sure and i was actually wrong, they absolutely did use them in battle.

english sources:

Goldsworthy, A. K. The Othismos, Myths and Heresies: The Nature of Hoplite Battle

Kagan, Donald, and Gregory Viggiano, Men of Bronze: Hoplite Warfare in Ancient Greece

Crowley, Jason. The Psychology of the Athenian Hoplite: The Culture of Combat in Classical Athens

Cartledge, P. Hoplites and Heroes: Sparta's Contribution to the Technique of Ancient Warfare

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non-english sources:

Andrea Frediani, Le grandi battaglie dell'antica Grecia

Paolo Taviani, Furor bellicus

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edit: typo