We don't have great evidence for this. There is a frequently cited passage from Suetonius' Lives of the Caesars regarding the emperor Augustus:

Festivals and holidays he celebrated lavishly as a rule, but sometimes only in a spirit of fun. On the Saturnalia, and at any other time when he took it into his head, he would now give gifts of clothing or gold and silver; again coins of every device, including old pieces of the kings and foreign money; another time nothing but hair cloth, sponges, pokers and tongs, and other such things under misleading names of double meaning.

Suetonius wrote around 120 years after Augustus died, so this habit of Augustus' must have been pretty well known in Rome at the time. Suetonius clearly lumps the "coins of every device" as a good gift along with clothing, gold, and silver. Suetonius also includes the fact that there were "old pieces of the kings", which by Augustus' time would have been hundreds of years old. Whether or not Augustus actively collected the coins for his own amusement is not known, although numismatists often credit him as the world's first coin collector.

We do have plenty of evidence that the Romans at least kept old coins for use as jewelry. One of the most famous Roman coins is the EID MAR aureus, struck by Marcus Brutus to commemorate the murder of Gaius Julius Caesar. These coins were largely destroyed shortly after Augustus defeated Brutus and his allies, when having one would have been an unwise political statement in Rome. Only 3 of these coins are known to still exist in gold, and one of them (in the collection of the British Museum) has a hole in it, indicating it was worn, probably as a necklace. Many other coins are found holed, indicating use as ornamentation as well.

Past that we don't have much direct evidence of collecting for its own sake in the ancient world. Certainly the busy trade networks brought coins of many kingdoms and cities all around the Mediterranean world. The ubiquitous Athenian tetradrachms have been found in digs all over Europe and the Near East. By Roman times, the earliest coins were already over 700 years old, and could have been viewed as artifacts in their own right. But we don't have surviving writings that specify collecting otherwise.

As far as collecting coins in general and Roman coins specifically, we have to go to the Renaissance. The poet Petrarch is the first documented collector of ancient coinage. He wrote that "Often there came to me in Rome a vinedigger, holding in his hands an ancient jewel or a golden Latin coin, sometimes scratched by the hard edge of a hoe, urging me either to buy it or to identify the heroic faces inscribed on them." This passage indicates that the diggers knew that these old pieces were worth something, at least to Petrarch.

In 1354, he presented Roman coins to Emperor Charles IV. "I presented him with some gold and silver coins, which I held very dear. They bore the effigies of some of our rulers, one of them, a most life-like head of Caesar Augustus, and were inscribed with exceedingly minute characters."

Around this time, royals and other rich people began to develop a taste for ancient coins, accompanying a general interest in all things Classical. It hasn't stopped since.

See:

Lives of the Caesars (Suetonius)

Letters on Familiar Matters XVII- XXIV. (Petrarch, translated by Aldo S. Bernardo)