It is solid inside (duh..) with a slight chewy texture, sorta like pig's kidney. If they are cooked correctly they should be easy to bite. Otherwise it will be rubbery. It doesn't have much taste itself, so whatever it is cooked with it will lend the flavor to the duck hearts. Chicken heart is pretty much the same, just smaller in size.
Most of the time when I have duck hearts it is at izakaya as yakitori. I have also tried the Chinese version which is marinated in soy and spices. Casa Mono used to have it on the menu but I never ordered it.