Payments were not only carried out through physical money transfer. In fact, it seems there used to be a pretty developed financial intermediation system (with varying space-time heterogeneity). Moreover, coinage was fairly widespread, diminishing the need for money transfers between places. And, when money had to be transferred, there might have been some fairly advanced system for it.
It seems loans were not only paid with money. Sometimes payments were made with land. For example, sometimes wealthy individuals loaned money to the state (this might have involved some initial money transfer). But then, as this article states:
in 200 the state gave citizens ager publicus as repayment of loans instead of money.
Land was of particular interest in the Roman economy. As the article states:
land was not only used for agricultural production, but also as a safe
investment. In societies where, in the absence of a complex banking system,
there are few possibilities of keeping money safe, buying real estate is the most
secure investment that can be made. Land, as opposed to many other
investments, does not lose its value and cannot be stolen as easily as precious
metals. Land was used as the security for other investments; therefore people
who did not own any land would find it difficult to invest in other enterprises.
Therefore, even those who were not interested in producing for the market may
have been willing to buy land.
Money transfers might have been performed also through other means. This article states:
Clearing between banks could be managed without any movement of coin by using rent payments, transfers of tax revenues, and so on.
Meanwhile, this one reads:
The inscription of a second-century Dacian loan says that the borrower will repay
whomever is holding the loan when it comes due (Sheldon, 1998, pp. 136-37). This
contract exemplifies the assignability of loans assumed by Livy. If the obligation went
directly from the borrower to the eventual holder of the loan without involving the
original lender, then the loan was negotiable as well. If so, then the loan might have
functioned as money, as notes obligatory did in 18th century England. The inscription
survives because it was immobile and largely indestructible; we do not know if there was
a wax copy that could circulate as paper contracts did later in England. In the absence of
documentary evidence for paper money, historians have tended to assume that all
transactions were made in coin. The existence of assignable loans at least raises the
possibility that there was “paper” money in use as well.
This article dedicates itself entirely to deal with the issue of money beyond coins. It states:
How did Cicero transfer the 3½ million sesterces he paid for his famous house on the Palatine (Fam. 5.6.2—this was by no means the largest property price we know of in the | classical city of Rome), at a time when Rome had practically no gold coinage? It seems singularly unlikely that his slaves counted out and loaded 3½ tons of silver coins and transported this cargo through the streets of Rome (not that Roman ideas of inconvenience were necessarily the same as ours). When a certain C. Albanius bought an estate from a certain C. Pilius for 11½ million sesterces (Cic. Att. 13.31.4), did he physically send him this sum in silver coins? Without much doubt, these were at least for the most part paper, or rather documentary, transactions (the crucial documents will have been waxed tablets). The commonest procedure for large property purchases in this period was probably the one casually alluded to by Cicero elsewhere: a Roman knight becomes enamoured of a certain property at Syracuse, and ‘nomina facit, negotium conficit’, ‘he provides the credits [or “bonds” ], completes the purchase’ (De off. 3.59). This practice is reflected in Cicero’s letters. And when in Pro Caecina Aebutius’ bid for a rural property being sold at auction is successful, he concludes the affair by (p.226) ‘promising money to the argentarius’, the banker (Caec. 16), and about that at least there was nothing in the least irregular: no one denied that the property had really been sold—the only question about the sale was whether Aebutius was acting for someone else. And you might buy in instalments: when Cicero bought out the share of the horti Cluviani that had gone to another legatee (Att. 13.46.3), he did so in three payments spread out over nearly a year (Att. 16.2.1), in effect taking a loan from the seller.
And later on:
One underlying reason for this growth in documentary transactions was clearly that in a Mediterranean-wide empire it was dangerous as well as inconvenient to send large sums of specie backwards and forwards over long distances. It was, of course, known to be risky to transport large sums of coin by sea, and both officials and private citizens probably tried to avoid it whenever they could. We have noticed that shipwrecked trading ships seldom seem to have carried many coins in high-classical times.
This article states:
Transfers of money were needed for investing in new voyages. Such transfers, which were effected by book entries, could pass through the hands of different intermediaries, publicans, ‘financiers to the aristocracy’, or members of the elites with interests or bonds of amicitia in the provinces.
This practice was not new to the ancient world. The Greeks also practiced it, as this article states:
Settlement of overseas commercial obligations by transporting moneys from Athens might have significantly reduced the supply of silver circulating in Attika. By guaranteeing payment of funds at far-off locations, the banks averted this drain and allowed customers to avoid the dangers and inconvenience inherent in transporting a large amount of coins or bullion. Thus when Stratokles, about to journey to the distant Black Sea, anticipated the need for currency there, he was able to leave his own money on loan in Athens and carry instead a bank guarantee of payment of principal and interest on 300 Kyzikene staters. This bank commitment was issued by Athens' largest bank, that of Pasion, in reliance on money remaining on deposit in the bank.64 Bank deposits thus effectively became ‘bank money’, enhancing the supply of coinage in circulation in Athens.
More about banking and financial intermediaries in some part of Roman history here.
Notice that the production of coins wasn't centralised. As this article states:
This rather straightforward history does not do justice to the coinage that must have illed the purses of most residents of empire, which was produced closer to home. Although denarii and (from the time of Augustus) aurei were certainly current everywhere, most coinage was produced in the provinces. ... During the Severan period as many as 400 cities issued coin.
So the supply of money was descentralised. I.e. not all money was produced in e.g. Rome.
This book might be also interesting, but I have no access to it. Further references that might be of interest are here.
Finally, naturally some (large) sums of money had to be transported. But it seems the Romans had designed advanced systems of transport. For instance, this aricle states:
how the actual system for transporting merchandise and coins over the desert worked. The cargoes would leave Alexandria, the big emporion on the Mediterranean, to be conveyed to Coptos on the Nile and from there overland to Berenike. As far as we know, merchants would borrow the money for their commercial expeditions from wealthy people willing to finance such trade, reaping huge profits from these loans. It is reasonable to imagine that these financers would also have provided the merchants with the coins to trade in India. At this point it is worth remembering that Roman coins found in India are mostly denarii or aurei —that is, types of coins of officially forbidden circulation in Roman Egypt. In fact, although the excavations at Berenike yielded Roman bronze coins and Ptolemaic tetradrachmai, not a single denarius or aureus was found. However, as the Roman denarii found in India did indeed arrive from Egypt, we are confronted with a seemingly insoluble contradiction. The answer to such a puzzle is, in my opinion, represented by the μαρσίππιον ἐσφραγίσμενος. The coins necessary for the trade with the Indians would be collected in sealed bags with a standard amount of coins (and, consequently, of standard weight). This would guarantee both the financer and the merchant: the financer would be sure that the traders could not open the bags and try to steal some coins, and the traders would be able to count the coins more quickly (they could be counted bag by bag, rather than one by one). Comparative examples for this hypothesis do exist. The first one is the so-called ‘tesoretto di Rimigliano’, discovered in 2002 off the shores of Italy. The ‘tesoretto’ (‘treasure’, or ‘hoard’) comes from a wreck in the Tyrrhenian Sea, and is composed of a group of silver coins (overall c.3,600, mainly antoniniani, but also a few denarii). Because of oxidation, the coins are still in the position they were in when the ship wrecked. They form a block of vaguely spherical shape, which led the editors to think that the coins were originally kept in a basket. The coins also appear to have been divided into smaller globular packages, small leather bags put into the basket. The small bags contain a standard amount of silver coins, split into groups of ten units, in order to facilitate the process of counting them. The ‘tesoretto’ has been interpreted as an example of the standard way of transporting coins on the Roman ships. Its structure can easily be compared with the one proposed for the μαρσίππια attested in the Berenike ostraka.