I don't think Walmart and Straight Talk are really "hiding" anything as Straight Talk uses all four of the "big 4" networks and it is known that they are an MVNO. An MVNO contracts with the big networks to buy service wholesale and then resells it to us, the consumer.
On their website, Walmart discloses the network most phones use right in the title description of the phone.
This phone, the Samsung Galaxy S III, has in its title line (CDMA-S) which means it will use the Sprint network.
As you scan the list of phones available on the Walmart website, you'll see that others will be denoted with either "CDMA-V" which use Verizon, or "GSM-A" that use AT&T.
So depending on what phone one chooses, their Straight Talk phone could use Verizon, Sprint, AT&T or even T-mobile.
Straight Talk is not "affiliated" with Sprint but they do contract to buy some bulk service from Sprint so that some of their phones will run on the Sprint network. Other Straight Talk Phones will use the Verizon network and others will use the AT&T network.
Straight Talk is actually "affiliated" with Mexico's América Móvil which is their parent company. So is Tracfone and Net10, other examples, a subsidiary of América Móvil.
This particular phone uses the Sprint network.