No. Human hermaphrodites are not actually both male and female. Genetically they will either be male or female, and thus will either produce sperm or eggs (ova), but never both.
People who are considered hermaphrodites acutually only have "ambiguous genitalia" usually due to hormonal imbalances in utero. For example, if a female fetus is exposed to testosterone in the uterus, she will develop ambiguous genitalia or if a male fetus is not exposed to enough testosterone in the uterus, he will develop ambiguous genitalia.
Chances are, no. In most cases a hermaphrodite is just merely a few organs of one or the other, and somewhat underdeveloped in one way or another. For the most part actual hermaphrodites are borderline non-existent, with a lot of cases being nothing more than just an oversized clitoris or something similar, and even if they had both genitalia, s/he likely wouldn't have the same internal plumbing, to the full extent, if at all.
So in short. No. Not likely. I highly doubt it, but I fully admit I am not an expert.