Do we know if these texts had much influence during their respective, somewhat-overlapping periods of prominence before and after the 1st and 2nd centuries? Note the Sibylline Books seem to have been destroyed in 83 BC and replaced a few years later, seemingly with newer contemporaneous prophecies. It seems the new version further had a "new copy" made ~12 BC when they were moved to the Temple of Apollo by Augustus.
The Sibylline Books (Latin: Libri Sibyllini) were a collection of oracular utterances, set out in Greek hexameters that, according to tradition, were purchased from a sibyl by the last king of Rome, Tarquinius Superbus; and were consulted at momentous crises through the history of the Republic and the Empire. Only fragments have survived, the rest being lost or deliberately destroyed.
The Sibylline Books should not be confused with the so-called Sibylline Oracles, twelve books of prophecies thought to be of Judaeo-Christian origin ...
The Roman Senate kept tight control over the Sibylline Books;[1] Sibylline Books were entrusted to the care of two patricians; after 367 BC ten custodians were appointed, five patricians and five plebeians, who were called the decemviri sacris faciundis; subsequently (probably in the time of Sulla) their number was increased to fifteen, the quindecimviri sacris faciundis. They were usually ex-consuls or ex-praetors. They held office for life, and were exempt from all other public duties. They had the responsibility of keeping the books in safety and secrecy. These officials, at the command of the Senate, consulted the Sibylline Books in order to discover not exact predictions of definite future events in the form of prophecy, but the religious observances necessary to avert extraordinary calamities and to expiate ominous prodigies (comets and earthquakes, showers of stones, plague, and the like). It was only the rites of expiation prescribed by the Sibylline Books, according to the interpretation of the oracle that were communicated to the public, and not the oracles themselves, which left ample opportunity for abuses.
.. The Sibylline Books motivated the construction of eight temples in ancient Rome, aside from those cults that have been interpreted as mediated by the Sibylline Books simply by the Greek nature of the deity.[2] Thus, one important effect of the Sibylline Books was their influence on applying Greek cult practice and Greek conceptions of deities to indigenous Roman religion, which was already indirectly influenced through Etruscan religion. As the Sibylline Books had been collected in Anatolia, in the neighborhood of Troy, they recognized the gods and goddesses and the rites observed there and helped introduce them into Roman state worship, a syncretic amalgamation of national deities with the corresponding deities of Greece, and a general modification of the Roman religion.
http://en.wikipedia.org/wiki/Sibylline_Books
2 - Eric M. Orlin (2002). Temples, Religion, and Politics in the Roman Republic; Ch. 3 "The Sibylline Books".
The influence of the books brought eastern gods such as Apollo, the "Great Mother" Cybele, and Ceres, as well as Greek pagan beliefs, into the Roman pagan religion.
Because the verses were written in Greek, the keepers would always be helped by two Greek translators. The books were destroyed when the Temple of Jupiter burned down in 83 BC. Because of this, the Roman Senate sent messengers in 76 BC to find similar prophecies and replace them. The prophecies were gathered especially from Troy, Erythrae, Samos island, 'Africa' (that is, modern Tunisia), and from Sicily and Tibur in Italy. After they brought the new collection to Rome, Roman priests separated what they thought was true, but threw others out of the collection.
The Roman Emperor Augustus in 12 BC moved them to the Temple of Apollo, when they were studied and a new copy was made. They stayed there until 405 AD. It is said that at that time, Stilicho, who fought for the teachings of Arianism, burnt them.
http://simple.wikipedia.org/wiki/Sibylline_Books
Modern scholars believe these Sibylline Books are not the same as the Sibylline Oracles that were often quoted by early Christian writers from the 2nd century through the 5th century AD. It is certain that when one Christian writer, Athenagoras of Athens, wrote A Plea for the Christians to Emperor Marcus Aurelius in around 176 AD, at a time when Christians were being punished by the pagan Roman Empire, he quoted word-for-word from these Oracles that are known today. Quoting them, along with writings by Homer and Hesiod, he wrote many times that "these books are all known to Caesar" (the Emperor). The Sibylline Books were still to be found in the Temple of Apollo at Rome at this time, so it is thought possible that at least some of these Sibylline Oracles were partly the same.
http://simple.wikipedia.org/wiki/Sibylline_Books
Relationship with the "Sibylline Oracles"
The Sibylline Oracles were quoted by the Roman-Jewish historian Josephus (late 1st century), as well as by numerous Christian writers of the second century, including Athenagoras of Athens who, in a letter addressed to Marcus Aurelius in ca. AD 176, quoted verbatim a section of the extant Oracles, in the midst of a lengthy series of other classical and pagan references such as Homer and Hesiod, stating several times that all these works should already be familiar to the Roman Emperor. Copies of the actual Sibylline Books (as reconstituted in 76 BC) were still in the Roman Temple at this time. The Oracles are nevertheless thought by modern scholars to be anonymous compilations that assumed their final form in the fifth century, after the Sibylline Books perished. They are a miscellaneous collection of Jewish and Christian portents of future disasters, that may illustrate the confusions about sibyls that were accumulating among Christians of Late Antiquity.[5]
http://en.wikipedia.org/wiki/Sibylline_Books
5 - http://www.sacred-texts.com/cla/sib/sib15.htm