Half siders, aka chimeras, are the result of two different genetic materials being fused together into one being.
Some are sterile, some are fertile, but it really depends on the genetics that are fused together. A bird could be male/male, male/female, or female/female. I can't say that it's been documented in parrots, but if a bird is born male/female halfsider (in theory, would require more research), then it may have been born with both sexual organs. One sex may be dominant over the other, so you might end up with a fertile bird after all, or the two might nullify each other and you have a sterile bird. In the same sex halfsiders, it may depend on how the bird was 'fused together' as to the fertility of the bird.
It's a subject that I'd love to know more about! But what I do know is that some halfsiders are in fact fertile and it was through breeding these birds that we were able to figure out that it is a genetic anomaly/defect and not a mutation.
http://en.wikipedia.org/wiki/Chimera_(genetics)
Of the species I can recall, chimeras/halfsiders have been seen in budgies, lovebirds, indian ringnecks and eclectus. This is speaking strictly of parrot species, not other bird/animal species where it's also been documented.
If we did more genetic/DNA testing on parrots, we might discover more chimeras that we didn't know about simply because the birds didn't show it visually.