The simplest answer is that the Iliad is not historical. It is an epic work of fiction that probably at least some basis in historical events. The ruins of Troy have been discovered, and there was a flourishing Mycenaean culture, but Paris, Helen, Hector, Achilles, and the rest are characters in a story. So they communicated the same way that characters from different countries do in modern movies or books - through the miracle of fiction.

However, if there had been a Trojan named Paris and a Mycenaean named Helen, they might have been able to communicate. Trade flourished in the Bronze Age in the western Mediterranean. Mycenaean ships sailed to Egypt, Crete, and Anatolia. Trade requires communication, so clearly somebody knew multiple languages.

No one knows for sure what language was spoken in Troy, but Luwian Hittite was the common language of the area. Luwian and Mycenaean were cousin languages.

Short answer: it is entirely possible that Mycenean Greek was spoken at Troy.

The occupiers of Troy I-V remain as much a question today as they were in 1900. The issue concerns the later layers, Troy VI through (especially) VIIa. They were either a Mycenaean settlement, or else all the Mycenaean artifacts and other evidence found in situ—and it is a lot—are Mycenaean imports and imitations (Manfred Korfmann sometimes suggested as much), and we do not know who the holders of the real estate were.

This is all the more important when you recall that the evidence we would most like to have was destroyed by two “disasters,” one ancient and one modern: the building of the great Temple of Athena in the 3rd century BC that cleared away all that remained of the central part of the citadel from this period; Schliemann’s "Great Trench," which buried whatever might be left over under tons of rubble after the erection of the Athena temple. Oops.

Charles Brian Rose thinks someday we, meaning some archeologist better equipped in the future, may be able to get at what lies under Schliemann’s big mess. But it was simply too dangerous to push the unstable, huge mound of rubble around when he and Korfmann were actively excavating at the site (1988-2012).

In any case, the best candidate based on evidence in situ argues for a Mycenaean Troy, as argued by Carl Blegen. I am in that camp until the physical evidence justifies a different conclusion.

Rose has also (quite honestly) written: “Especially striking is the absence of Hittite pottery at the site…” (Archaeology of Greek and Roman Troy p. 35). I tend to find that convincing.

And yes, maybe under Schliemann’s horrific pile there may reside tons of Hittite artifacts, but until we can prove that, and dig it up, we must argue from what has been found at the site. That’s how historians are supposed to work.

The main pieces of the Mycenaean evidence include a Mycenaean seal that serves as a counterpoint to Korfmann's Hittite hieroglyphic Luwian seal; also see the Mycenaean pottery remains in Manfred Korfmann’s Guidebook, which could be nothing but Mycenaean, p. 111.

Manfred Korfmann can also be a bit slippery, however. He writes: “…[this] speaks for the identification of the site of the Hittite vassal state Wilusa.” That does not quite say, does it, that the site was occupied by Hittites, but only that at one point whoever occupied the site were regarded as a vassal state by the Hittites, who had lots of vassal states that went in and out of “vassaldom” as fortunes changed, and allies shifted? Could that have happened to Mycenaean Troy? Yes, of course.

Is the identification of Wilusa in the Hittite documents with the site of Troy certain? Far from it. That is what Hittite scholars (and I am not one) tell me.

In other words, we cannot assume that a person from Mycenae and person from Troy in the era of VIIa wouldn't have spoken the same or a common language.