The short answer is: The people of South Africa own the bones, according to the scientists from the expedition into the Rising Star Cave, 30 miles northwest of Johannesburg. The University of the Witwatersrand curates the fossils on behalf of the people of South Africa.
The Cradle of Humankind, the nickname given to the area of South Africa that yielded many hominin fossils during the 20th century, is also a UNESCO (United Nations Educational, Scientific and Cultural Organization) World Heritage Site and as such is governed by its rules. The South African Heritage Resources Association, or SAHRA, works with UNESCO and the South African government to protect the entire site.
In July 1998, at a meeting of the Permanent Council of the UNESCO-affiliated International Association for the Study of Human Paleontology in Sun City, South Africa, the issue of taking hominid fossils from their country of origin was discussed. Afterward, a resolution was passed, unanimously approved by the Permanent Council and adopted by the Assembly of the Association.
The resolution included two parts, one supporting the use of replicas of hominid fossils for public display and at museums to promote awareness about human evolution, and a second strongly recommending that original hominid fossils not be transported beyond the boundaries of the country of origin, "unless there are compelling scientific reasons which must include the demonstration that the proposed investigations cannot proceed in the forseeable future in the country of origin." Representatives from 20 countries, including both South Africa and the United States, signed the resolution.
For the analysis of Homo naledi, Lee Berger's team invited dozens of experts to come to the Cradle of Humankind in South Africa to examine the fossils up close.