Some do, some don't. My father is Hades, so I can raise and control the dead, manipulate shadows. I can't do much with the earth, but my sister, she can pull metal and gems right out. [There's a note of affection as he thinks of his Roman half-sister.] Percy, one of Poseidon's sons, he can control water, and talk to horses, breathe underwater. [There is some affection here, too, but he shrugs it away.] Not all demigods are born with abilities, but the more powerful the godly parent, the more likely the kid will be powerful too.