The price of sequencing continues to decrease as the technology evolves. I have already seen claims of under $1,000 for a full human genome. I haven't looked carefully into those claims, but I think we are around there. In some years full genomes will be so cheap to sequence that it will be routine. I want to buy one of those small Oxford Nanopore MinION sequencers in the future. I'll use it like a pokedex.
There is a theoretical future in which full-genome sequencing is performed exclusively by large companies, hospitals, and governments, and the data is stored by them and they can access it.
But the technologies are becoming quite accessible. Unless regulations are introduced to force people to give up their genetic data, which I don't think is so likely, there will be ways for us to get our sequences without the sequences being stored by a third party. I also think that there will be FOSS tools for us to run our own analyses.