Infectious diseases like AIDS, malaria and tuberculosis have all sorts of terrible symptoms. Left untreated or at their most severe, they can cause death. With that in mind, sweating seems like, well, nothing to sweat about.
But this uncomfortable symptom can certainly make people with these diseases feel much worse. Night sweats are all quite common with each of these ailments. Patients will wake up drenched in sweat as a result of a severe hot flash, often associated with fever and infection from these diseases.
Now, these are all infectious diseases that can be transmitted from person to person. But not to fear, shaking someone's sweaty palm is not going to spread any of these diseases.