tl;dr - Overall, very rare.

The most important factor to keep in mind is reason to learn. For the vast, vast majority of medieval Christians, there was simply no reason to learn Arabic. They might know "pagan" Saracens existed, but that would be about it.

As you mention, OP, there were a few cases where Latin Christians would indeed have exposure to the Arabic-speaking world. The biggest group is probably not diplomats but merchants. There were European (mainly Italian) expat merchant communities in quite a few Mediterranean ports--Tunis, Cairo and Alexandria have been the focus of important scholarly work in the last few years. Merchants would likely (almost certainly) not have had formal instruction in Arabic, but it's reasonable to assume they learned enough to get along.

One case I'm not familiar with is the Crusader states in the Near East of the high Middle Ages. There is not a ton of evidence for mixing among the very small Latin populations with the original residents. But again, practicality may have dictated that people learn a few phrases of each other's language. Robert of Ketton was an English scholar who traveled extensively through the Near East, learning Arabic quite well along the way. He would eventually settle in Spain and translate the Qur'an into Latin around 1143.

As for formal instruction, there were a couple instances! You mention the Muslim communities on Sicily. It's true that some did remain after the Norman conquest, and indeed, the Normans enjoyed the extra taxes they could levy and didn't try to convert the Muslim population en masse. However, on one level they practiced a policy that led to the gradual isolation of the Muslim community, its eventual mass deportation to a colony on the Italian mainland, and their eventual wholesale expulsion. Alex Metcalfe, probably the foremost scholar on Islamic Sicily, speculates that there were a few bilingual Arabic-Romance communities, but (a) says the evidence is inconclusive and (b) says there is no evidence for what kind of bilingualism--meaning, people used one language daily but knew a few phrases in another; people moved in between languages regularly; etc.

The formal opportunity, though, came at the Norman royal court.

In the 12th century, the Norman administration in Sicily actually adopted Arabic as a prestige language! They were very plugged into the trans-Mediterranean world, drawing a lot more of their influence on governance/bureaucracy from Fatamid Egypt than from their Latin counterparts (twelfth century Europe is juuuust figuring out the business of governing instead of violent marauding). The Norman administration issued bilingual charters, with Arabic used as a show of power locally and also for practical purposes in diplomatic communications. Along with governance, the Norman court was also the site of major scholarly efforts at translating Arabic original and ancient Greek survival texts into Latin. Yes, there were Muslim scholars as well as Christians working at court, fluent in Latin as well as Arabic and probably a handful of vernaculars. /u/caffarelli - many of these "palace Saracens" were eunuchs!

At the same time the Normans were encouraging scholarly efforts at translation in Sicily, the courts of Iberia were fostering the same intellectual life. So you had Latin Christians learning Arabic formally there, too. There are a few hints of formal studia arabica (Arabic schools) established. The traditional modern understanding is that these were intended to teach future missionaries Arabic for conversion efforts, but more recent work has sort of undermined that idea--the Dominican and some Franciscan friars in North Africa, for example, were mainly there to minister to the expat European populations, not to convert the local Muslim residents. So if the studia arabica were just meant for people who would be working on translating texts, they'd have even more limited access.

Overall, there were a few formal opportunities for a very few, very educated, very elite Latin Christians to learn Latin. In the Mediterranean zone, exposure to Muslim communities may well have allowed more people the chance to pick up anywhere from a few phrases to practical bilingualism, maybe the equivalent of an English-speaking American with a telenovela addiction. International merchants and dedicated travelers (...mainly merchants, a few missionaries), still a pretty insignificant number overall, would likely have known enough Arabic to get by. But for the average European Christian, especially outside the Mediterranean zone, there was neither the opportunity nor the reason to learn Arabic.

Sources of Note: Alex Metcalfe, The Muslims of Medieval Italy; Jeremy Johns, Arabic Administration in Norman Sicily: The Royal Diwan; Georg Christ, Trading Conflicts. Venetian Merchants and Mamluk Officials in Late Medieval Alexandria; Robin Vose, Dominicans, Muslims, and Jews in the Medieval Crown of Aragon