You actually have 3 choices with Straight Talk, AT&T, Verizon or T-Mobile networks. The sim card program ( bring your own phone)only works on either T-Mobile or AT&T, if you want the Verizon network you have to buy the phone from them.
I have been on Straight Talk (BYOP) AT&T network for a few months now, works great everywhere except my house, only 1 signal bar, we're in a AT&T dead zone (very weak signal. So far I have no complaints.