In the Middle Ages, conjoined twins (or any so-called "monstrous birth") were usually interpreted as ill omens from God, presaging judgment or disaster of some kind.

This was the typical response to anomalous events (e.g. comets), especially ones that seemed to run "against nature" (e.g. snow in summer). These events were not seen as random, but as specific signs often directed at a particular town or city (in response to sinfulness, typically). This interpretation reigned well through most of the Middle Ages, and beyond it into the seventeenth century.

Isidore of Seville's Etymologiae (Book 11.3.4):

Some portents seem to have been created as indications of future events, for God sometimes wants to indicate what is to come through some defects in newborns, and also through dreams and oracles, by which he may foreshadow and indicate future calamity for certain peoples or individuals...

This was not a uniquely medieval view. It was inherited almost wholesale from ancient Greek and Roman traditions, although nuanced by Christian tradition. Classical tradition would have viewed conjoined twins as a simple aberration of the natural order, but there was significant pushback from some Christian thinkers (St. Augustine in particular) on this idea. The Augustinian impulse was to see everything (even the anomalous and the bizarre) as a direct and intentional creation of God. Augustine himself even questioned the idea that such anomalies always presaged disaster, but most medieval Europeans clung firmly to the idea of ill omens.

Still, even though deformed births presaged something grave, and would tend to elicit panic and horror amongst the local population, they were viewed as a creation of God and treated accordingly. Thus, we see accounts of deformed infants being baptized and cared for until death, which often came within a few days. The response of the local authorities could really vary from place to place, and was a matter of local decision. Here's a few accounts to demonstrate possible reactions. Notice the negotiating of reverence, care, horror, and curiosity.

Guibert of Nogent, 12th century French abbot, on the omens foreshadowing the 1116 revolt of Laon:

[Some descriptions of a comet, evil spirits, and flames appearing the air] . . . Some days before, a baby was born who was double down to the buttocks; that is, he had two heads and two bodies right down to the loins, each with its own arms; double above, he was single below. After he was baptized, he lived three days. In short, many portents were seen to occur which left no doubt that they presaged the great disaster which followed.

Giovanni Villani, 1317, Florence:

In the said year, in January . . . there was born in Tarraio di Valdarno di sopra a boy with two bodies; he was bought to Florence and lived more than twenty days. Then he died in the Florentine hospital of Santa Maria della Scala, first one body and then the other. And when it was proposed to bring him alive to then priors, as a wonder, they refused to allow him in the palace, fearing and suspecting such a monster, which according to the ancients signifies future harm wherever it is born."

Petrarch, 1342, on the same event:

"Some friends in Florence sent a picture [of the child] to us in France, where we were staying... and a huge crowd of people came just to see it. I was seven years old when I saw the image in the hands of my father. When I asked what it was, he told me, showing it to me, and ordered me to remember it and tell the story to my (as he said then) my sons. And I will indeed tell it to my nephews."

As this last anecdote reveals, these births were quite rare and attracted a fair amount of attention even after the death of the child. There was often decent documentation of these events, and particularly at the close of the Middle Ages (c. 1500s) there was a flurry of printed tracts and leaflets that depicted anomalies like the child in Florence.

The response to a deformed child was usually fast, since the children often did not live long. The parents, midwife, and parish priest had to determine whether the child was human, one person or two, and whether to engage in acts of penitence to stave off disaster. Unfortunately, I don't know of any cases of conjoined twins surviving into adulthood. If any other commenters have examples of this, I'd really love to know.

Source: Lorraine Daston, Wonder and the Order of Nature, 1150-1750 (New York: Zone Books, 1998), 21-66.