The field of genetics is relatively a new one within the science world. As within every industry, the discovery of new things and new processes leads to legal issues, including intellectual property rights and patents. The field of genetics has been no different.
As genes, specific sequences of DNA, began to be discovered, scientists started requesting patents on "their" genes. Previous to 2001, some of these discoveries (identification of a gene's sequence alone) were granted patents. In 2001 though, questions were raised about patents of genes and some demanded that the bar be raised for patents. Their argument was that these patents will limit the use of basic genetic information, and may inhibit or slow biomedical research. They had a solid argument in my opinion.
The concern was well received, and the grounds for a patent were modified. The current guidelines state that "identification of a gene's sequence alone is not patentable, but that a gene isolated from its natural state may be patentable if the applicants can demonstrate 'specific, substantial and credible utility' for the discovery" (from Genome.gov). These guidelines have allowed for the worldwide sharing of human genomes, in order to increase the data available and allow the field to advance further.
Still, if your genome is tested, do you want it available to just anyone? Or is that information your own property? Currently it is available to anyone, although names are not attached to genomes. I don't honestly see an issue with it, but someday if we are using our genomes to allow pharmacy to give us 'taylor-made' drugs, I could see an issue with a database of genomes being available online. People may take advantage of it to illegally obtain drugs they otherwise could not obtain. Ultimately, the industry is still just taking its first baby steps in my opinion, so time will tell how these issues are resolved.