In fact it was just one of Italic alphabets; they all were developed in about the 7th century BC. Another famous legend tell us that Romulus and Remus were taught literacy in Gabii, a town in Latium inhabited by Greek colonists - so actually the first Roman kings did not know writing at all.
Archaeologists have found about 25 different varieties of the Italic alphabets in the country. The most widespread was the Etruscan script, or more correctly the Etruscan set of alphabets (about 8). While the Roman script was borrowed directly from Greek, most of the rest Italic writing was transferred from Greek via Etruscan. This can be said about Oscan, Umbrian, Volscian alphabets. The Venetic script also derived from one of the north Etruscan alphabets, the same about Lepontic and Rhaetic languages.
This is explained by the important Etruscan influence on all tribes
in Italy at the time when writing spread over the country. Later, when
Etruscan power was crushed, nations of Italy more likely accepted the Roman
script. Since the 3rd century BC, about half of all Italic inscriptions,
written in Oscan, Umbrian and other languages, were made in Roman. Soon
all small varieties of Italic alphabets were forgotten, and the last inscription
found in Etruscan dates from the 1st century AD.
The original Etruscan alphabet consisted of 26 letters and was written from the right (which is important, because most Indo-Europeans write from the left). It was taken directly from Western Greek, but the majority of symbols are reversed. Several letters are original and unique, they are believed to have been invented in Italy.
Oscan, Umbrian, Volscian, Venetic and other alphabets of Indo-European tribes accepted the Etruscan writing, but it appeared not quite convenient for Indo-European languages: for example, in Etruscan there was no sound [o], which coincided with [u], so Italic nations had to follow this rule and to write [u] instead of [o]. Nowadays it is therefore hard to say if there was [o] in Umbrian. In Etruscan fricatives could be only voiceless, there was no [b], [d], [g], and this also hardens the Italic script transliteration.
A strange sample of writing was found on the island of Lemnos in the Aegean Sea. Written in an alphabet close to Etruscan, it was made in the unknown language, also related somehow to Etruscan and Rhaetic. Was it an Etruscan colony in the Aegean, or a relative nation, remained unclear. This Lemnos Stele is the only example of Italic inscriptions beyond Italy itself.
Languages which used Italic alphabets: Italic
(Oscan, Umbrian, Volscian, Picene), Venetic,
Illyrian (Messapic), Celtic
(Lepontic); non-Indo-European (Etruscan, Rhaetic, Sicanian, Lemnian).