We only have one decent set of written sources to answer this question: the letters between Trajan and Pliny the Younger. Pliny was based in Nicomedia and governed the province of Bithynia; in what is now north-west Turkey. They show that the line between 'government' and 'private' investment was often a thin one, and that the government and private investors often worked together on construction projects. Quite often, 'private' investors were either Roman or local senators. Generally speaking, the bigger the project, the more the imperial government paid for it.

There was nothing in ancient Rome which we would call a company; there were a near-limitless number of businesses which we would call 'cottage industries', and there were some guilds but they were limited in scope. If you wanted a large quantity of wine in Pompeii, for example, you wouldn't contact a guild because, as far as we can tell, there was no guild of wineries in Pompeii - you'd have to shop around the various wine sellers. If you needed a large number of goods shipped to the 12th legion, you wouldn't contact a guild, you'd contact individual merchants called sutlers and make some arrangements. There was a class of organisation called a 'collegio' which is normally translated as a guild, and they did sometimes fund small building projects (quite often temples to Fortuna), but the Roman emperors generally tried to suppress their influence because of how they were thought to cause trouble. An exchange of letters between Pliny the Younger and Trajan show that even a guild of 150 people was deemed too dangerous:

Pliny to Trajan:

...consider whether you think a guild of firemen, of about 150 men, should be instituted. I will take care that no one who is not a genuine fireman should be admitted, and that the guild should not misapply the charter granted to it, and there would be no difficulty in keeping an eye on so small a body.

This letter tells us that guilds existed, and that they had charters, or at least something very close to one. Pliny was clearly expecting Trajan to object to this on the basis that guilds could get out of control, which he did:

Trajan to Pliny:

You have conceived the idea that a guild of firemen might be formed in Nicomedia on the model of various others already existing. But it is to be remembered that your province of Bithynia, and especially city states like Nicomedia, are the prey of factions. Whatever name we may give to those who form an association, and whatever the reason of the association may be, they will soon degenerate into secret societies.

In short, Trajan did not like guilds, or indeed any large organisations. He expected them to be corrupt and underhanded, and to cause trouble. This issue came up again when Pliny explained how an aqueduct had been poorly half-built by the local government (an elected senate) of Nicomedia and then abandoned at the cost of 3,329,000 sesterces. Pliny suggested that the project be taken over by the Roman administration, but Trajan had an additional task in mind:

Trajan to Pliny:

Steps must certainly be taken to provide the city of Nicomedia with a water-supply, and I have every confidence that you will undertake the duty with all necessary diligence. But I swear that it is also part of your diligent duty to find out who is to blame for the waste of such sums of money by the people of Nicomedia on their aqueducts, and whether or not there has been any serving of private interests in thus beginning and then abandoning the works. See that you bring to my knowledge whatever you may find out.

Trajan is essentially wondering if any Nicomedian senators run construction businesses, and have used their position as senators to enrich themselves through building a phony aqueduct. Whether this distrust of private investment was unique to Trajan or whether it was more widely shared by the Roman elite, we do not have the evidence to confidently say.

The thinness of the line between government and private investors in paying for building projects is demonstrated in a failed joint venture to build a theatre.

Pliny to Trajan:

The theatre at Nicaea, lord, the greater part of which has already been constructed, though it is still incomplete, has already cost more than ten million sesterces, - so at least I am told, for the accounts have not been made out, - and I am afraid the money has been thrown away. For the building has sunk, and there are great gaping crevices to be seen, either because the ground is soft and damp, or owing to the brittleness and crumbling character of the stone, and so it is worth consideration whether it should be finished or abandoned, or even pulled down. For the props and buttresses by which it is shored up seem to me to be more costly than strength-giving. Many parts of this theatre were promised by private persons, as for example the galleries and porticos above the pit, but all these are postponed now that the work, which had to be finished first, has come to a stop.

Trajan to Pliny:

You will be best able to judge and determine what ought to be done at the present time in the matter of the theatre which the people of Nicaea have begun to build. It will be enough for me to be informed of the plan you adopt. Do not trouble, moreover, to call on the private individuals to build the portions they promised until the theatre is erected, for they made those promises...

This theatre, a significant portion of which was paid for by private investors, is being taken over by Pliny. Trajan has, however, said that Pliny can call upon those investors to pay up in order to repair and complete the theatre.

What Pliny's letters show is that building projects were paid for through a multitude of methods. At a local level, the local government paid for construction through taxation, often supplemented by private investment, which might have included guilds. However, we can see that Pliny was at liberty to intervene as he saw fit, take over any of these projects himself, and fund them from the imperial treasury if needed - especially if he suspected financial mismanagement on the part of private investors. A temple or amphitheatre could therefore have been paid for by a variety of sources.