You're severely underestimating the amount of multilingual people that would have existed. Merchants, scholars, or diplomats would likely have known several languages and/or had access to people who could translate for them. In some cases a bridge language may have been used (ex: an American and German who both knew French, but not each other's language).

I'd assume it's much more likely the Persians knew Greek than the Greeks knew Persian. There were Greek colonies/cities on the Anatolian coast within territory the Persians claimed. Persia exerting influence over the Ionian coast and mainland Greek city-states supporting the Ionians is what sparked the Persian invasion of Greece in the first place.

It's a fallacy to think of the ancient world as disconnected entities. There was a huge amount of trade and diplomacy between different cultures even well before the Persian/Greek wars. The leaders and scholars knew about each other's customs, traditions, culture, and language quite well.

I'm gonna go out on a limb and assume you're a native English speaker, because even today, there are an awful lot of multilingual people in the world, with probably over half knowing at least 2 languages. The dominance of the English language worldwide has led to a lot of anglophones not only not learning a second language, but resultantly not developing the skills for picking up languages. It's pretty well known that picking up new languages gets easier the more languages you already know, which actually means that ancient diplomats probably had it easier. You see, nowadays we generally have this idea that languages are independent and relatively rigid things, but before modern communications technology local dialects sprang up quite easily, and they could vary a huge amount even within the same language. This means that anyone who travelled around in the pre-modern world would essentially have to be able to integrate different languages into their knowledge base much more regularly than someone would have to today. The nature of dialects also means that languages in contact would often develop pidgins that could be used as a way to learn the other language by people who knew one. The standardization of languages really put an end to that.

More Greeks fought on the side of the Persians than they did on the side of the Greeks. These were cities that were vassals to Persia (Like in Asia Minor) or they were cities on the Greek mainland who simply thought that the Persians had a better chance and wanted to be on the winning side.

A few Greeks even famously changed sides midbattle (Thessalians).

Many Greek mercenaries from Athens and Sparta and other cities fought for the Persians both before and after the Peloponnesian wars. (see Xenophon)

All these Greek Vassals would have people speaking both languages by necessity.

There were even people in the Persian court who were Greek exiles.

I would argue that learning languages was easier for many people then, especially in centers of trade like a Greek city-state. As the other commenter mentioned, there were an abundance of multilingual people because of the large exposure to different languages.

There were ambassadors between countries whose main job was this very thing.

First of all, there was plenty of international trade going on, so a surprising number of people probably spoke multiple languages. Merchants in particular, as well as bureaucrats responsible for customs duties and tax collection probably spoke multiple languages.

Secondly, most rulers probably had multiple scribes who were each literate in multiple languages as well.

There were also probably common languages as well, most likely originated in the big empires. So, a lot of people probably spoke Greek, Latin, Egyptian, or Persian to some extent.

It's not exactly at the same period, but to give you a picture in the Viking Age at the beginning of the medieval period, the vikings were trading with Constantinople (and I mean a big trade route), and sometimes with Spain and Italy. They also had spies in about all the royal court of the major countries of Europe and again in Constantinople. So yeah, people managed to communicate.

Plus, you must not forget that people at the time had slaves. So one might capture someone and sell him to, let's say an intellectual, looking to learn the language.

the persian´s had geek subjects, if need be some of those could be asked to translate

King Leonidas: "We've been sharing our culture with you all morning."

Seriously though, much like in the modern day, there were multilingual people. And major city states and neighboring empires would keep trained citizens, slaves, and freedmen who were versed in multiple languages of important trading partners, regional powers, and potential enemies.

Specifically for the case you are referring to, the Persian king Xerxes had an exiled Spartan king Demaratus as an advisor in his court. Demaratus accompanied Xerxes I on his invasion of Greece in 480 BC and is alleged to have warned Xerxes not to underestimate the Spartans before the Battle of Thermopylae.

It probably wasn’t any harder back then to be honest, the human brain hasn’t fundamentally changed enough in this amount of time. The internet may be helpful, but I would still say the easiest way to learn a language is to be exposed to it at a young age.

Children of merchants and diplomats likely shadowed their parents and that type of profession would simply give them the chance to practice frequently.

As others have pointed out, there were plenty of polylingual people back then, and the Persians in particular would have had access to lots of subjects who were fluent in Greek, because they were Greek.

Another thing to recognize is that there were probably at least two languages that were widely understood because they belonged to people who engaged in lots of long-distance trade across the Mediterranean basin. One of those was, of course, Greek, since several wealthy Greek city-states (e.g., Athens, Corinth, Syracuse) did a lot of maritime trade. The other would have been Punic (Phoenician), and the home cities for that language were under Persian rule throughout the period.

The official language of the Persian Empire was Aramaic, a Semitic language. The Persians sent letters Aramaic written in cuneiform to Sparta. Aramaic was the linga franca of the Middle East from about 700 BCE until about 700 CE, when another Semitic language, Arabic, took over.

Old Persian was similar in many ways to Ancient Greek, but the languages were not mutually comprehensible.

The Persian empire took control of Anatolia which had many Greek Colonies like Pergamon, Rhodes, Sardes, etc. They had subjects which spoke both Persian and Greek.

I don't know if this could help but, as far as I remember, regarding the Persian wars historian Herodotus sometimes is mentioning the use of interpreters [uses the word ἑρμηνεύς], while at other instances he places persons of different nations speaking directly. This underlines the possibility that he implies that they were speaking with the same language.

Trade and to even an certain sense global trade existed. Hundreds of years of mixed stories and intelligences.

Also Persia being a super-nation state would have hired,Merced out, or raised, Greek speakers.

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Mostly by translation. Keep in mind leaders rarely met face to face, and most communication was thus written, usually by scribes rather than the leaders themselves. A leader with a lot of resources could probably find a scribe that speaks Persian and can write in the various forms of Greek common at the time, or vice versa. Also, leaders tended to be multilingual themselves.

There were greek mercenaries in the Persian army, not to mention the other Greek populations the Persians were overlords of. Not all Greeks were against the Persians. Also, after the war, themistokles was later ostracized and served the Persians for the rest of his life. https://en.m.wikipedia.org/wiki/Themistocles

Go to the sub section titled "Greeks in exile to the achamaned empire" I'm sure you could find more info for other Greeks serving the Persians before the war following that source

There's a super-famous example of this very thing that linguists often cite as a clue to the divergence of languages from a common root. I've forgotten the specifics, but it comes down to two kings/war-leaders/chieftains taking an oath of friendship. One of the big dudes spoke French and the other German. So the French speaker delivered his oath in German so the German-speaking troops could hear and understand him, and the German-speaking dude dutifully followed up by repeating HIS oath in French so the French-speaking troops would know what he was promising. I was like, cool, that's neat, but the significance of this is it's the first recorded instance of French and German having diverged sufficiently from their common predecessor language as to be unintelligible to someone who didn't speak French/German.

Finding translators has been a real challenge in international relations for as long as there have BEEN international relations. The problem is that there's not a huge amount of material in the historic record as to how this was handled.

We know that Sacagawea was enough of an expert in Native Sign Language (as it existed at the time) that she was able to teach a lot of it to the Lewis & Clark party.

The pre-Renaissance reclamation of ancient Greek-language texts on natural science, literature, philosophy, art, mathematics, engineering, and astronomy relied on European monks finding translators who spoke Greek, Latin, and Arabic, and who were also scholarly enough to be able to explain the contents to the monks doing the transcribing. Legend has it this is how we got the symbol for "zero"; the monk was puzzled by the concept, so the translator obligingly recommended that he just draw a little hole in the middle of the equation.

Too, there's the example of La Malinche (a title, not her name, which has been lost) making great use of her multilingual skills to help Cortez destroy the Aztecs.

This particular story isn't told too often, but in the aftermath of the Second World War, displaced persons all over Europe took to the roads to try to walk home. As the roads got larger, the displaced found themselves sharing with the victorious military. At select crossroads, the Allies stationed multilingual Europeans who had survived the mayhem to talk to the displaced people, getting their names, home locations, and stories to help with what must have been a world-class traffic jam.

I guess, if contemporary civilization collapses, anyone who is multilingual is gonna be very, very prized.

My Uncle Joe ran a neighborhood grocery store in Chester Pa. back in the 30's and 40's. He had no formal education beyond High School, but could converse in German, Polish, Yiddish, just by doing business with his customers. I'm pretty sure the same held true with so many diverse tradesman across ancient borders.

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This of Europe today... Back then, those who could travel knew far more than today's travel able Europeans (whom are already known for know 3/4 languages just cos Europe)