Prisoners and slaves were taken by both sides throughout the 12th century before Hattin, but we don’t really know a lot about what happened to them. Muslim slaves taken by the crusaders were probably mostly used as domestic servants in the cities, but slaves were also important for construction work and rural agricultural work. Slaves were used to build the Templar castle at Safed, which is probably the most well-known example of Muslim slave labour.

Before Hattin, there wasn’t really any government-led attempts to get Muslim slaves back. They could run away, and if they could make it back to territory ruled by Muslims, they would be free. Otherwise, freeing slaves was an individual effort - Usama ibn Munqidh, an ambassador from Damascus who often visited the crusader kingdom, spent his own money to ransom as many slaves/prisoners as he could.

Likewise, there was no official attempt to retrieve Christian captives. It was up to individuals to ransom enslaved prisoners, if they wanted to, and if they could afford it. Generally the crusaders had no idea what happened to Christian prisoners, or how to get them back. There is a whole series of epic poems about Christian prisoners taken during the First Crusade (“les chétifs” - the captives), and there were legends about them - Zengi, the sultan of Mosul, was supposed to be the son of a captive Christian woman. Usama mentions Christian slaves sometimes as well. In one of his stories, a Muslim ruler took a Christian slave as his wife, but years later the woman managed to escape back to crusader territory. (Sexual slavery was a likely outcome for prisoners on both sides.)

Hattin was the first time the crusaders had to deal with the problem of a huge number of politically important nobles being taken prisoner.

“After Hattin the plight of captives shifted from an individual to a national or even international predicament." (Friedman, pg. 88)

They appealed to the pope and other European leaders to help them pay the ransoms, and around the time of the Third Crusade, the “Order of the Trinity” was founded, with the specific goal of raising money to ransom crusader prisoners. The Hospitallers and Templars and the other military orders could also use their financial resources to ransom prisoners.

The most notable person captive at Hattin was King Guy. He was taken to Damascus and eventually freed, in exchange for the city of Ascalon. He’s depicted being paraded in front of the walls of Jerusalem riding backwards on a donkey in Kingdom of Heaven, but I don’t think there’s any historical basis for that. Similar things did happen though. Reginald of Sidon was brought to Sidon to convince the garrison to surrender - he told them to surrender in Arabic so Saladin could hear, but then he told them in French not to surrender. William of Montferrat was brought to Tyre, where his son Conrad was defending the city, and Saladin threatened to kill him unless Conrad surrendered - but Conrad responded by shooting a crossbow bolt at them, almost killing his father himself.

So we know what happened to some of them - they were ransomed, in exchange for money, for entire cities, or for Muslim prisoners of similar rank. For less-significant prisoners, they might be imprisoned for a much long period, or they would be enslaved.

Most likely, however, prisoners who were enslaved, both Muslim and Christian, weren’t enslaved for life. After Hattin, an ransoming network developed so they could be ransomed eventually. Depending on where they were, it was also relatively easy to flee back to their nearby homes.

If they couldn’t be ransomed and they couldn’t escape, then unfortunately no one knows what really happened to them. Some of them must have converted to Islam/Christianity; sometimes in crusader sources, there are disputes about what to do with a crusader prisoner who returns home but had converted to Islam while imprisoned. There are also occasions in crusader sources where Muslim slaves try to convert to Christianity and their crusader owners refuse (because legally Christians weren’t allowed to own Christian slaves).

So, in summary, high-ranking people were normally ransomed relatively quickly. Lower-ranking people could be ransomed as well, but it might take longer, and in the meantime they might be enslaved. Unfortunately we don’t really know what life would be like for an individual enslaved person because no one bothered to write that down, so we just have to guess from other clues.

The best source for this is:

Yvonne Friedman, Encounter Between Enemies: Captivity and Ransom in the Latin Kingdom of Jerusalem (Brill, 2002).

Another good place to look is Benjamin Z. Kedar, "Multidirectional conversion in the Frankish Levant", in Franks, Muslims, and Oriental Christians in the Latin Levant (Ashgate, 2006), for the more specific issue of prisoners converting the other side's religion.