Polybius was the son of a Greek politician and had started on a political career himself when at the end of the Third Macedonian War he - along with many other disloyal Greeks - was brought to Italy as a hostage. Taking these hostages ensured the loyalty of the Greek cities.

For Polybius this was the end of his political career, but this event did put him on the path he is known for today. He decided to write a history of Rome, to determine how Rome succeeded where the Greeks failed: "Can any one be so indifferent or idle as not to care to know by what means, and under what kind of polity, almost the whole inhabited world was conquered and brought under the dominion of the single city of Rome, and that too within a period of not quite fifty-three years?" (1.1)

Polybius stayed in Rome and befriended the quite powerful Cornelius family. As an educated Greek, he became a tutor to Publius Cornelius Scipio Aemilianus (usually just called Scipio Aemilianus or - later - Scipio Africanus-minor). Scipio Aemilianus paternal grandfather was Lucius Aemilius Paullus, the consul who was killed at the Battle of Cannae in 216 B.C. Scipio Aemilianus adoptive grandfather was Publius Cornelius Scipio Africanus, who defeated Hannibal at the Battle of Zama. Scipio Aemilianus himself was the one who defeated Carthage in the Third Punic War.

This is one reason why Polybius is quite detailed: he had very close ties to the direct family of some of the most important figures of the Second and Third Punic Wars. Oral traditions get more inaccurate as time goes by, but Polybius was able to hear it all told within a few decades of everything happening. For some of the later history, such as the Third Punic War, Polybius was actually present. Polybius often joined Scipio Aemilianus on his travels and commands; he was present at the destruction of Carthage by Scipio Aemilianus.

Furthermore, Polybius had enough freedom to do the things historians do. He could talk to other Romans and even foreign emissaries which were sent to Rome. He could read books and reports written by other historians. He even was free to do some travelling - for instance he visited the Alps to try and find out which route Hannibal took exactly.

The fact that Polybius was writing about events just prior to and in his own time, and the fact that he made an effort to check for himself the things he was told make his reports pretty accurate (as far as reports in ancient history go).

That said, I wouldn't go so far as to say he described events "from both sides". He reports from a decidedly Roman point of view (except perhaps where the Greeks are concerned). He continuously describes the Carthaginians as untrustworthy and immoral. Within Polybius' Roman view there are some indications he tries to make the ancestors of Scipio Aemilianus look good. For instance, if Polybius is to be believed the disaster at Cannae was fully consul Terentius Varro's fault, whereas consul Aemilius Paullus tried to prevent it. This is very likely not true. (I wrote about that here)

As for the battle descriptions you ask about: this is really down to what was reported orally or in written reports. The accuracy of the reports is often discussed by historians, sometimes in minute detail. The exact location of battles is something which can be discussed at length (archaeology can help here, but in the case of Cannae there has been discussion on which side of the river the battle was fought and even how much the river has changed its course since antiquity).

Troop numbers are often exaggerated by ancient historians; as you suspect historians adjust these numbers based on what else is known (how many legions were present, the average size of legions and their auxiliaries, etc.). Generally troop numbers for the Romans are more accurate (or at least more easily verifiable) than other troop numbers because of the standard organisation of the Roman army. Casualty numbers again have to be considered critically. Often the number of reported casualties and the number of reported survivors varies between (and sometimes even within) accounts. Historians deal with this as best they can, by giving ranges or by trying to find other evidence (archaeological or epigraphical) to support one number or the other.

edit: spelling