Some scholars like to date the Arab contention of Sicily a little farther back than the 120 years you mentioned; however, the other side of the coin is that many scholars including Byzantine primary sources consider those to have been mostly raids rather than full on campaigns for control of the island. These raids go back to the 7th century and see an increase in sustainability once the Umayyad Caliphate had firm control of North Africa giving them access to the ports of Carthage. The Muslims did pose a major threat to the Byzantines in Sicily as early as the 8th century but many scholars tend to agree that the real attempts of expansion began in the early to mid-9th century.

I wanted to mention the constant raids of Sicily from an early point because whether you believe it was the Rashidun, the Umayyad or the Abbasid Caliphate that tried to conquer Sicily, all of these raids/battles devasted the lands and negatively impacted the population of the island. In 827 Mazzara del Vallo was captured and we can take a quick fast-forward to 902 where almost the entirety of the island of Sicily was firmly under the control of the Aghlabids. If you are unclear about the Aghlabids, they were a ruling family that controlled Sicily and the region known as Ifriqiya, which includes modern day Tunisia, and bits of Algeria as well as Libiya. The Emirs of the Aghlabid dynasty controlled these regions on behalf of the Abbasid Caliph.

Here is where it really gets complex: because of all the warring and all of the raiding, the population shifted. You saw many Latins and Greeks straight up migrate out of Sicily and into Calabria which caused the population of Calabria to migrate into Lucania, or you saw them take refuge in the higher grounds in the north eastern portion of the island, resulting in many lands to be poorly or completely uninhabited (sources & regions vary). Many Berber nomads were brought to settle in these lands and because this island was in the center of the Mediterranean many people from Al-Andalus (Islamic Iberia) migrated for opportunity, as well as other North Africans and merchants from the Middle East and this caused a sort of mixed society on the island.

I purposely used the phrase sort of¸ and the reason being is because it wasn’t totally mixed. The further you went in one direction the more of a specific religion you saw. So, if you were to start from East to West you would start out in a mostly inhabited region of Latin and Greek Christians and the further West you went, somewhere in the middle it would be more mixed and if you got to the West, it was now mainly Sunni Muslims. This isn’t to say that the Latins and Greeks didn’t convert, they most certainly did as time went on for different reasons; however, it seems they were more resistant about changing religion than they were about changing their culture. This was evident once the Normans got to Malta where they were shocked to learn that the Greek Christians of the island spoke Arabic.

The conquest of Sicily by the Normans was done in separate campaigns at different times. It began with Roger and Robert invading in 1061 by crossing from Calabria into the island and attacking Messina. The conquest of Sicily by the Normans would not be complete until the beginning of 1091.

The Normans expelled the Muslim government but did not expel the Muslims of the island. The Muslim population of the island was way too important for many reasons, especially trade, which explains why the Normans did not receive any help from the likes of Amalfi. The Muslims on the island are mentioned throughout the rules of Roger I, Roger II, William I and William II, meaning that up until the end of the 12th century there was a significant population of Muslims in Sicily without efforts from the Norman overlords to try and convert them; in fact, the Muslims received more pressure from the locals to convert rather than the Norman administration.

At the very end of the 12th century the whole of southern Italy which would be known as the Kingdom of Sicily fell under the control of the German Emperors. It is reported by their metrics that in 1250 the island of Sicily was overwhelmingly majority Latin Christians.

I’m sure conversion was a factor considering Sicily was under Islamic rule for over 2 centuries; however, the major shifts in religions on the island were more because of the migrations in and out of the island.

Now to briefly touch on your question regarding the Shia:

The sources don’t mention much in regards to the Shia Muslims because although the Fatimids with the assistance of the Berbers overthrew the Aghlabids, Sunni scholarship flourished to such a degree in Sicily that the Shia just simply had no time to make any sort of impact. Many scholars came to and out of Sicily during the Islamic control of Sicily, in fact too many to even name.

Arabic Administration in Norman Sicily by Jeremy Johns

The Age of Robert Guiscard by Graham A. Loud

The Muslims of Medieval Italy (New Edinburgh Islamic Surverys) by Alex Metcalfe

The Normans in the South by John Julius Norwich

The Latin Church in Norman Italy by G.A. Loud

Medieval Sicily, al-Andalus, and the Maghrib by Nicola Carpentieri

Storia dei musulmani di Sicilia by Michele Amari

Il monachesimo greco in Sicilia by C.D. Fonseca