Can you make sperm and eggs from the same person using in vitro gametogenesis?
At present (early 2017) the answer is No. 

The trick to make spermatid like cells (ie they are not real sperm cells and need to be injected into an oocyte to accomplish fertilization) from ES and iPS in a dish was just achieved last year http://www.sciencemag.org/news/2016/02/new-method-grows-sperm-dish

The trick to make oocyctes from stem cells is not as complete and still requires an lab grown ovary to complete the process. http://www.sciencemag.org/news/2016/10/mouse-egg-cells-made-entirely-lab-give-rise-healthy-offspring.

The methods above still use ES cell of the appropriate gender. Female ES cells to make oocytes. Male ES cells to make sperm.  No info on female ES cells being driven to make sperm or even spermatid like cells.

That said... we have this paper http://news.nationalgeographic.com/news/2004/04/0421_040421_whoneedsmales_2.html, where a mouse is genetically engineered to produce oocytes that have imprinting patterns similar to a sperm. This genetically modified oocyte is then forced to fuse with a normal oocyte. Success rate was very low. 371 fused oocytes, 10 live births, 1 surviving to adulthood.