This is a good question, but unfortunately the information available does not make for a very satisfactory answer. We depend mostly on literary evidence, and the sources are maddeningly vague on this point. I think that it is fairly safe to assume that the Homeric example, at least in part, was still the norm in most Greek city-states in the Classical period. In Homer, the possessions of the dead were pooled and distributed by the king/leader/hero. There are of course exceptions to this, usually by exceptional characters. From the pile is taken the dekate (the 10th share for the gods)(Iliad 10.460 e.g.) and special awards for aristeia, bravery or battlefield accomplishment. Then the rest is divided up. This distributive concept is very "Mycenaean" in its flavor, and it wouldn't surprise me if such a system was a hold-over memory from the Greek Bronze Age. We don't know for sure, of course.

In the Classical world, our evidence is broad but unfocused. We know that allied contingents in the Persian wars had pre-arranged agreements for how booty should be distributed. So at Plataia in 479, Diodorus Siculus says (11.33) that the distribution of booty was based on the number of soldiers in each contingent (this suggests it was pooled ahead of time to get an accurate "count," not just looted at random by individuals). After Salamis, Herodotos says the Greeks piled up all the spoils, sent first fruits (dekate) to Delphi, and then divided the rest. He does not bother to say how it was done (VIII.120-22).

I personally think that the idea of individual booty was somewhat stigmatized in the Classical Greek world. The hoplites fought as a unified whole, for their city, shoulder to shoulder, and there was little room for individual accomplishment or reward, despite the stories of Homeric heroes they heard around the fire at night. This changed from polis to polis, and through time. The soldiers on the march in Xenophon's anabasis might have behaved different concerning booty than an Athenian hoplite in the Peloponnesian War. We know that the Athenians brought back the spoils of the defeated Spartans from Pylos in 425 and set up the shields (at least) in the Stoa Poikile, a public monument (Aristophanes Knights 843ff). Xenophon even plainly sets it out in Anabasis, though this was probably a special circumstance: "Whenever the army remained in camp and rested, individuals were permitted to go out after plunder, and in that case kept what they got; but whenever the entire army set out, if an individual went off by himself and got anything, it was decreed to be public property. (Xen. An. 6.6.2) Also it should be noted that the "plunder" was not necessarily from dead soldiers, but included farmhouses, hapless peasants, and chicken coops.

After the disastrous Athenian invasion of Sicily, the Syracusan soldiers apparently tried to smuggle Athenian prisoners out as their personal possessions instead of bringing them to the collecting points (Thucydides 7.85). If a soldier was motivated (and able!) to smuggle live human beings, I think we can safely assume that other objects also went home with them. What happened to that sweet Lakedaimonian hunting knife after that is anyone's guess.

Speaking of Spartans: they had official "booty-sellers" (laphuropolai) who were responsible for disposing of collected plunder immediately. I do not know whether the proceeds were collective, or were then distributed to the soldiers.