There are a lot of possible configurations of interstate currency issues, and each would have its own distinct solution. Here I'm going over a few that are familiar from a broadly Classical Mediterranean context, but a medievalist or early modernist might be able to provide a bit more.

1: The Countermark

The countermark is an incredibly simple concept: one state that receives coins of another can, instead of going through the process of recasting or overstriking them, instead punch onto the coins a local design in order to validate them as legal tender. Countermarks might be necessary for a number of reasons. For example, it might be used as an assurance of purity. In many contexts, though, it was also used as a means of permitting coins made on a certain weight-standard to retain their denominational value when used in a place that typically used another, even if it contained more or less metal.

To digress a bit, a weight-standard is, as the name suggests, the standard system of weights used in a particular place. To use a Greek example, the Attic weight-standard used in Classical Athens was based on the 17.2g tetradrachm (4-drachma coin), the Aeginetan weight-standard used in Aegina was based on the 12.2g didrachm (2-drachma coin), and the Cistophoric weight-standard used in Hellenistic Pergamon was based on a 12.9g tetradrachm. Cistophoric tetradrachms were worth a tetradrachm within regions that accepted the Cistophoric weight-standard, but could not circulate as such in regions using the Attic standard, which had become commonplace thanks to both the extent of Athenian imperialism and its use by the Macedonian kingdom under Alexander. Instead, at best they were worth their weight in metal. Such 'closed' currency systems, which could not be easily converted to the Attic standard, were known as 'epichoric'. But such systems don't exist in total vacuums, and often do need to accept external currency for internal use.

Some good examples of countermarking can be found on pages 119-126 of Peter Thonemann's The Hellenistic World using Coins as Sources. Coins of the city of Side, based on a roughly 16.5g tetradrachm, were countermarked for use in Sardis, with a punch bearing a chariot design and the letters ΣΑΡ, and in Seleucid Syria, with a punch bearing the standard Seleucid symbol of an anchor. As an aside, just to show how hard these punches go in (as the coins were struck without prior heating), on the latter example you can quite clearly see on the reverse a significant amount of flattening where the punch went in on the other side. This piece from late 3rd century BC Byzantium is also quite a fun one, with a ship's prow and the letters ΒΥ (in local script). While the original Seleucid coin (minted at least 50 years earlier) was based on the Attic standard, Byzantium at this time used a 13g tetradrachm, so counterstriking the Seleucid coin affirmed that its value in the city would be its denominational value of 4 drachmae rather than its value in weight of metal, which on the Byzantine standard would be 5.3 drachmae.

Countermarks could also be used to indicate the continued validity of an issue in spite of political changes. You can find pictures here of a bronze issue by the city of Tripolis in the last years of Nero's rule, which was a rather unfortunate time to be minting civic coinages given the whole Year of the Four Emperors thing that ended up happening in AD 69. Nearly all of them have the words 'IMP NERO' removed, some by scratching them off, but many by adding a countermark. Most known examples have 'IMP GA' (short for Galba), but some read 'IMP OTH' (for Otho) and some 'IMP VES' (short for Vespasian). Sadly, it seems that Tripolis never fell under Vitellius' orbit, and so we don't have all four of the Four Emperors depicted there.

Irrespective of specific purpose, countermarks are just a simple, recognisable sign affirming a piece's value. Probably the best examples come from Qing China, where Spanish and Mexican silver dollars were often countermarked as a guarantee of validity. This coin of Charles IV is a good example of the level of countermarking typical of Spanish coins in China, but some can be pretty much FUBAR – this coin of Ferdinand VII is basically only identifiable as such because the year of minting survives. American trade dollars were also heavily countermarked, and you can see a collection of 20 of them here.

2: Weight

It's easy to forget that money and coins aren't the same thing. Money is really just any sort of relatively non-perishable store of wealth, whereas coins are specifically metal items bearing designs. While coining metal does expedite a lot of exchange, particularly at small levels, bullion is an entirely valid way of storing and moving money as well. A lot of early money was just some form of cast metal – Sparta's iron rods, Etruscan cast bronze and so on – and in some places, this inheritance from simple cast bullion is reflected in the early coins. Rome, for example, originally produced cast bronze asses (worth 1/16th of a denarius, or rather vice versa) that were worth their weight in metal, which meant producing these pretty gigantic 1-ounce cast bronze lumps depicting a head of Janus and a ship's prow (hence the Roman expression 'Januses or prows?' in place of our 'heads or tails?'). I have handled one, and they are pretty hefty, at about 3 finger-widths in diameter, and on average maybe a finger width thick. But to return from the digression, bullion was evidently always an acceptable medium of exchange so long as it could be weighed. Uncoined metal can often occur alongside coins in hoards, and plenty of silver and gold objects dedicated at sanctuaries have their weights given in equivalent quantity of coins – sometimes of multiple weight-standards.

At the same time, there's no reason why you can't simply weigh the coins and use their weight in gold or silver, converted to local standards without having to alter the coins themselves. Basically all the Roman coins found in India were minted before AD 64, when Nero reformed the Roman coinage, and it seems that most of the Republican pieces in particular arrived (based on contextual evidence) after Trajan's debasement of the silver coinage in 107, suggesting that the Romans were specifically using old, high-purity silver that had fallen out of internal circulation, but still had export value in the form of its weight in metal (Howgego 1995 p. 104). The pattern of coin distribution in Caucasian Iberia (located around modern-day Georgia), through which the northern, landward branch of the 'Silk Road' ran, is broadly similar. Again to take the Qing example, Spanish silver seems to have circulated in South China with its value originally reckoned by weight, and conveniently one silver dollar was just about 2/3 of a tael. To take a different classical example, in Plutarch's Life of Agesilaos, the titular Spartan king is said to have declared that King Artaxerxes drew him from Persia back to Greece 'with ten thousand archers', alluding to Ionian gold and silver coins which were stamped with images of a figure drawing a bow, a reference to the shipments of money to Athens and Thebes that the Persians had provided.

One point to note here is that it tends to be that precious metal coins could circulate because of their bullion value, even if their face value was not always accepted, but bronze coins were typically fiduciary (that is, they derived their value from the authority of the issuer, not their weight of metal) and tended to be restricted to their minting cities. In some cases, bronze pieces might circulate within a group of participating cities, such as in Ionia or, on a larger scale, East Asian states like China, Korea and Japan that had limited access to precious metals, but for most interstate exchange, bronze could be exchanged – at a fee – for precious metal. In 2nd century BC Thebes, 137 drachms' worth of bronze coins could get you 110 drachms' worth of silver (a whopping 25% commission for the exchanger!) which shows how constrained in use bronze could be compared with precious metal. (Thonemann 2016 pp. 139-41)

3: Don't Transport Coins

One key dynamic here that's been forgotten is that there's rarely good reason, as a merchant, to bring a load of coins with you instead of goods. Basic economics suggests that it is far better, where possible, to carry cargo both ways, not just money. As a budding merchant, you might buy a cargo with money (likely loaned) from your home city, travel to a destination, sell those goods for local money, use all of that local money to buy new cargo, and sell that cargo in the next city, hopefully covering at least some of your debts while giving you what you need to repeat the process. In this series of exchanges, money (coins or otherwise) doesn't actually have to leave the city it is issued in, and instead facilitates exchange within them. Basically one of the only exceptions was Athens, according (via Howgego) to Xenophon in Ways and Means:

...at most other ports merchants are compelled to ship a return cargo, because the local currency has no circulation in other states; but at Athens they have the opportunity of exchanging their cargo and exporting very many classes of goods that are in demand, or, if they do not want to ship a return cargo of goods, it is sound business to export silver; for, wherever they sell it, they are sure to make a profit on the capital invested.