Rome did not chiefly use divination to learn the future in general terms. It used it to ask whether a specific proposed action had divine approval, which made it an instrument of public decision-making with real procedural force.
The auspices were the standard form. A magistrate about to take public action observed the sky, or watched the feeding of sacred chickens, and reported whether the signs permitted him to proceed. An adverse report could halt an assembly or postpone business, and the announcement of unfavourable signs was open to obvious political use, Roman writers say so themselves. Extispicy, the inspection of the entrails of sacrificial animals, was practised with the help of haruspices, specialists drawn from an Etruscan tradition that Rome treated as foreign expertise and consulted for centuries without ever quite absorbing.
Prodigies were reported to the Senate: lightning striking public buildings, deformed births, statues that sweated or moved, showers of stones. The Senate decided whether each was a matter of public concern and, if so, ordered the appropriate expiation. The Sibylline Books, a collection of oracles kept under guard and consulted only on the Senate's instruction, were reserved for the gravest cases, and the remedies drawn from them sometimes brought new gods and new rites into the city.
Educated Romans argued about all of this. Cicero, himself an augur, set out a thorough case against the reliability of divination in a dialogue on the subject while defending the institution's public utility. That combination, scepticism about the mechanism, insistence on the practice, is characteristic, and it is a further reason not to read Roman rites as expressions of private conviction.