What happens if someone is allergic to a medicine they need to be taking... can.they be desensitized to it so they can take it?

Yes can be done: Allergic reaction ( to any thing ) is complex process , several factors & different mechanisms are involved . Allergy - like penicillin which will give anaphylatic shock (and death ) is caused by type-i mast cell / ige reaction , is being done for certain drugs , by specialized centers ( or offices ), and is not a routine procedure.