The short answer is that they had a fundamentally different method of collecting and expending revenue than most modern societies, but they certainly had political disagreements about spending priorities.

The first and most important thing to realize is that the Roman republic was continually at war for most of its history, and it was very successful in that enterprise. Loot (money, slaves, etc.) and other resources (i.e. gold and silver mines in the provinces) acquired from conquest, and the subsequent taxation of conquered territories, was a huge supplement to the state's income. Significantly, the income from conquered provinces was so great that after 167 B.C. no taxes were levied against Roman citizens in Italy. Prior to the point where foreign gains made domestic taxes unnecessary, taxes were primarily levied based on a percentage of the value of property and wealth (not on income). Wealth was determined as part of a census every five years, and persons in the poorest census class (proletarii) paid very little in taxes because they owned almost nothing besides the labor of their children (proles). Tax rates increased in times of national emergency or particularly dangerous wars. Taxes were sometimes also collected in kind (particularly grain).

Another important feature was the way in which the state collected taxes in the provinces, which also affected planning. Romans used a system sometimes referred to as "tax farming." The state auctioned off the right to collect taxes in a given province for a certain number of years at a certain rate to the highest bidder. The bidders (called publicani) paid up front for the right to collect taxes. The publicani then collected the actual taxes from the province and kept whatever they got. From the perspective of the Roman state, this was a great deal; they got a known sum in advance and didn't have to actually collect the taxes. From the standpoint of the publicani, this was also potentially a good system because if they could extract more than they paid they could pocket the difference (although they also took a risk of back to back bad harvests, etc.). From the perspective of the provincials, this could be a very poor system because the publicani were incentivized to squeeze just as much as they possibly could out of the provinces, and in many cases squeezed more than the nominal rate they were permitted to tax. This was in part why tax collectors had a poor reputation in the ancient world.

Another thing to understand is that many projects that we today would think of as "public works" were handled differently. Roads were typically built by the army, either as part of their campaigns, or as off-season work when not campaigning. As such, the labor costs of building roads were frequently just a harvesting of the sunk costs of military pay. Other public buildings (such as the building of theaters, public temples, or water systems) were often financed by wealthy individuals and given to the public, as a method of enhancing their public prestige and political careers.

When you add up these differences, the picture is in some ways very different than our own. The state often knew its likely or actual revenue in advance, so it determined its budget based on the cash on hand and the military campaigns underway. Common people paid little in tax in the early republic and no tax in the late republic, which affected the psychology of public spending. So the Roman government knew about how much money it had on hand annually, which set the budget, but, like us, there were certainly conflicts about how to use public money.

An easy example is the grain supply. The population of Rome quickly exceeded the ability of the surrounding areas to supply grain, so grain had to be imported. For most of the history of the republic, the grain was bought and sold normally. The Gracchan grain law of 123BC provided subsidized (but not free) grain to citizens as part of a broader, extremely controversial, platform of popular reforms. The popular reforms (the land reforms specifically) were so controversial that they actually sparked an outbreak of political violence that set the stage for the first civil war a few decades later. For the following half-century the price of subsidized grain (i.e. the amount of the subsidy) in Rome became a political football, especially when external events raised grain prices. There were also occasional non-recurring distributions of free grain. In 58 BC, the state began regularly offering free grain to some number of citizens at significant cost. After that point, the political debate focused on how many people would be allowed to collect free grain, and that debate persisted well into the imperial period.

Additionally there were often bitter disagreements about how to provide for the retirement of soldiers returning to Rome after conquests. A general returning at the head of a victorious army had to convince the Roman government to ratify campaign bonuses/retirement settlements for his soldiers (often a cash payment and some land). This proved contentious as the soldiers' retirements were often entirely dependent on the political savvy of their general, and a general who got his soldiers a good deal enhanced his prestige and political base. This led opponents of a given general to attempt to reduce or interfere with his settlement on return from campaign. This turned the soon-to-be veterans into a political football, and (crucially) aligned the interests of the returning soldiers more strongly with their general (who was trying to get them paid) than with the state (which in some cases was trying to stiff them/reduce their pay). This kind of spending dispute (and the resulting perverse incentives) was one of the contributing factors in the civil wars that ultimately ended the republic.

This is a very interesting question because it explores the definitions of private and public property in the Roman republic. Which was a hotly debated topic through most of the republican period, and eventually led to the formation of the Empire. But the short answer is that the Roman state had no real budget and most public works were payed for by wealthy members of the senate.

When most of us think of the Roman republic we think of it being ruled by a benevolent representative elected senate. However, the senate was technically an advisory committee who could be elected from the wealthiest and oldest families in Rome as determined during the census by officials called censors. The two members of the senate who shared the highest elected positions were consuls who led the armies of the Republic against its enemies. Because the senate is technically an advisory group the 'state' of Rome didn't have a national budget in the way that we would expect. For the most part any public works were payed for by wealthy individual members of the senate, who would usually emblazon their name across the facade of the building to remind people who payed for it.

Wealth in the Roman world was primarily defined by land holdings, and as the empire began to grow, the military played a key role in the characterizing how money was earned and spent. For the early portions of the republican era each class of Roman citizen, besides the lowest caste, had to provide men and arms for military service depending on their station. The lower classes would have provided themselves with a spear and armor, while a wealthy senator could field a small army. As these forces conquered new lands,questions began to be raised as to who owned these new acquisitions. With loot, the solution was easy the commander would distribute the money and wealth after a campaign to his soldiers in what was known as a donative. In the case of land this was more difficult, it raised a question, did the land acquired by roman conquest belong to the state, or to the wealthy who payed the largest amount for the armies to conquer it. This question gets very complex and gets entangled with roman law over land, agriculture, and with the roman grain dole that would take much longer than a Reddit comment to explain. But if you are interested in knowing the political drama behind it that reads like a game of thrones plot please read the chapters on the Gracchi in Mary Beard's book SPQR. Suffice it to say that despite legislation to stop them the upper classes of Roman society managed to find loopholes and take the newly acquired lands into their holdings.

Keith Hopkins lists these profits of empire as the single greatest factor in the accumulation of wealth by the Roman elite. He describes the process in three steps,

the conquest of territory brought profits of empire,

the wealthy recipients used this to purchase land in Italy from small rural farmers, forming large estates worked by slaves,

with this new capital, they could afford to field armies or fund the armies of their patrons, which allowed further expansion and renewed profits of empire continuing the cycle.

Its this cycle that leads to the rise of men like Julius Caesar and Pompey the great. Who could field enormous armies with their own wealth who were loyal to them over the senate because they signed the paychecks. This system led to years of civil war and eventually the rise of Augustus and Rome as an empire.

At this point the coffers of the Emperor became synonymous with those of the state and it was with their personal wealth that roads, aqueducts ,and the Colosseum were built.

TL;DR: The lines between public and private wealth were fuzzy in Rome, but for the most part things were funded by the private coffers of wealthy senators and eventually the Emperor.

I hope this is a sufficient answer I'll be glad to clarify things or fix any formatting if I need to.

Sources:

Beard, Mary. 2015. Spqr. First edition ed. New York: Liveright Publishing Corporation, a division of W.W.Norton & Company.

Hopkins, Keith. 1978. Conquerors and Slaves. Sociological Studies in Roman History. 1. publ. ed. Vol. 1. Cambridge [u.a.]: Cambridge Univ. Pr.

Mary T. Boatwright, Daniel J. Gargola, and Richard J.A. Talbert. 2004. The Romans : From Village to Empire. GB: Oxford University Press.

Edit: haven't had my coffee yet, fixing typos will be an ongoing process

I’m anxiously awaiting an answer to this question here. If I can also ride the coat tails and ask a secondary question when people start to reply: What are resources that discuss how currency was managed (by the Republic and the Empire) as well? I know at times certain coinage was revalued. What were the circumstances surrounding that and also the results?

Weren’t mid-level bureaucrats like aediles expected to foot the bill for infrastructure and cultural projects in their respective jurisdictions? I also remember reading that urban policing was outsourced to guilds and gangs as well, since armed soldiers weren’t allowed inside the city...wondering how true that is.