The earliest Roman coinage dates from the late 4th Century, B.C., probably involved in the process of the construction of the Via Appia and contacts with the Greek cities of Campania, which had had coinage for a little while already. Before that the Romans, like the rest of the Italic peoples, with the occasional and irregular exception of some Etruscan cities, evaluated property and taxes by the weight of bronze (aes rude). The first Roman coinage we know of was minted in 326 and consisted of little bronze coins of pretty low value bearing the Greek inscription ΡΩΜΑΙΩΝ, "of the Romans" (inscriptions on coins are regularly in the genitive in Greek, since the purpose is to indicate to whom the coin belongs). The Greek inscription is noteworthy--it's not as though the Romans had no written language or anything, the inscription is in Greek because the earliest Roman coins aren't really Roman, they're Greek. These coins were minted by the Neapolitans to commemorate the alliance between Naples and Rome, and they are distinctly Neopolitan--they were minted at Naples and they're identical to Neopolitan coins of the same period, it's just that the ΝΕΑΠΟΛΙΤΩΝ has been replaced with ΡΩΜΑΙΩΝ. There were also silver didrachmae minted from Naples and indistinguishable from Neapolitan coins of the same period, just that they say ROMANO (notice that this inscription is in Latin for some reason and is also in the dative instead of the genitive). But these coins were pretty much totally insignificant to the Roman economy and probably didn't circulate much--new coinage was not minted until probably around the time of the Pyrrhic War. Even then in the period between the First Punic War and the Pyrrhic War coinage wasn't really used, but instead these cast bars, or spits, much like the early Greek obols and probably influenced by cities in Magna Graecia, were used (and are often mistakenly called aes signatum, even though that term just means any minted coin)--earlier the aes rude might also take the form of these funny little tokens shaped like double-headed axes or seashells or whatever, which are not really coinage proper. Around this time the aes grave was also introduced, a strange case-bronze (casting was rare in ancient coinage) rounded token weighing a certain amount of bronze. They came in a few denominations but the basic one was the as bearing Janus' head and weighing an astonishing pound of bronze--but the images on these were very irregular and not at all standardized until around 240, though the earliest ones appear around the time of the Pyrrhic War. You may have a replica of an Uncia, the smallest sub-denomination of the early as, bearing the image of Roma or Bellona, but I don't think so--those coins look like this, and the helmet crest of your figure is different, as is the appearance of the helmet, and it's missing the pellets around the rim. Like I said, though, these coins were extremely irregular and even after the faces appearing on the coins were standardized the style of the coins wasn't really, so it's possible--although if it is supposed to be one of the aes grave it's been erroneously cast to look like it's silver, not bronze. Around the time of the Punic Wars the silver coinage replaced the ROMANO inscription with ROMA in the nominative, and shortly thereafter, under the financial pressure of dealing with Hannibal, the coinage system collapsed and was replaced with a new silver coin, the denarius, in 211. The image you're showing is certainly not a coin later than this period--the first denarii are much more complicated and expertly crafted than the image you've got there, bearing a helmeted Roma on the obverse and the Dioscuri riding horses on the reverse. If those are genuine replicas of a real coin then they probably come from the period predating the denarius (and are probably miscast as silver replicas of coinage of a period that was almost entirely in bronze), given the relatively crude image and the total lack of any inscription. If that's true, identifying it is kind of a pain in the ass and definitely something that only a trained numismatist, rather than a political historian like myself, could answer--coinage of the 3rd Century is incredibly unstandardized when it exists at all, and there are loads of little variations in same sort of coins

You might want to crosspost this to /r/ancientcoins, this is right up their ally.