According to herpetologists who are completely in the know that I've spoken with - a handful of people in the United States (that is to say they know of TWO people) are breeding them but are reluctant to report it due to a fear of their animals being confiscated.
They are also legal to export with no limit on quantity out of Indonesia if the eggs are incubated and captive hatched - this is considered captive bred on their part.
Some good reading:
3.3 Illegal trade
Indonesia: As noted in 3.1, above, a substantial proportion of eggs harvested in West Papua is incubated in captivity and the hatchlings are exported into the pet trade. Being hatched from wildcollected eggs, this trade is in a grey zone of legal/illegal trade under Indonesian legislation. However, in the absence of clear documentation of sustainable management practices, this trade is generally considered illegal and such traded animals are confiscated. Samedi and Iskandar (2000) and Samedi et al. (2002) noted illegal trade of Carettochelys from Merauke and Timika, West Papua, to Makasar (Ujung Pandang)(Sulawesi), Jakarta and Surabaya (Java) and Denpasar (Bali) and exported onwards to China and Singapore. In 1999, local prices were about 10,000–15,000 Rp (USD 1.12–1.69) per hatchling in Senggo and Atsy, where eggs were incubated, about 30,000 Rp (USD 3.37) per hatchling after transport to Merauke, and 60,000–70,000 Rp (USD 6.74–7.87) each in Surabaya (Maturbongs, 1999). In February 2004, three separate shipments of Carettochelys juveniles were intercepted and confiscated at Jakarta’s Soekarno-Hatta International Airport. The shipments comprised 100, 390, and 309 Carettochelys, respectively, and were all destined for Japan (Fidrus 2004).
Peoples Republic of China and Hong Kong S.A.R.: The Endangered Species Import and Export Management Office of the People’s Republic of China’s Identification Manual for common Turtles and Tortoises (2002) lists Carettochelys as ‘rarely seen in trade’. A survey of 5 pet shops in Hong Kong SAR and one pet market in Guangzhou, China, carried out between 30 October 2000 and 13 October 2001 observed 354 individuals of Carettochelys offered for sale as pets. The species was not encountered during surveys of food markets during the same period. The shops and markets were surveyed twice during every winter month and once during each summer month; thus, the number observed is a minimum number, not an approximation of total annual volume (Ades 2002).
3.4 Actual or potential trade impacts
The general consensus is that recently intensified egg collection, to a large part driven by the aim to supply hatchlings to the international pet trade, is presenting a clear threat to the survival of the species in West Papua (Maturbongs 1999) and is also affecting the exploitation and conservation status of the species in Papua New Guinea (Rhodin and Genorupa, 2000). Inclusion of Carettochelys in CITES Appendix II is primarily intended to facilitate and reinforce existing export restrictions in the three Range States, by providing an international dimension to the species’ protection. This will greatly increase opportunities to reduce illegal trade in the species by imposing trade control measures in the importing countries. At present, only the Lacey Act of the United States of America provides comparable complementary protection. Inclusion of the species in CITES Appendix II should not compromise the subsistence utilization of the species by native communities throughout its range. This could facilitate limited, sustainable harvest and trade with associated socio-economic benefits by and for these communities.
4.3.1 International trade
Once exported from their respective Range State, Carettochelys specimens are subject to national regulations pertaining to species trade, customs, and quarantine measures when entering the importing country. In the United States, the protected status in the Range States makes the species subject to the provisions of the Lacey Act. In most countries, regulations require compliance with the International Air Transport Association (IATA) regulations concerning the shipping of live animals, as a condition for acceptance or transit passage through airports (IATA Live Animals Regulations, Chapters 1and 2). In addition, most airlines require shipping of live turtles to comply with the IATA regulations (IATA Live Animals Regulations, Appendix A).
and so on and so on...
So basically they're illegal under a bi-law created in the early 1900's to prevent animals being imported that could potentially be hazardous to the environment if released, which has since been revised to include all sorts of stuff and carries a maximum penalty of a $20,000 fine or 5 years imprisonment.