Verizon and Sprint uses CDMA technology for their cell service. AT&T and T-Mobile uses GSM. CDMA does not allow surf and talk at the same time. However, GSM does. Some phone makers did allow some of their phones to surf and talk while on a CDMA cell service. But that only worked when that particular phone was using two antennas at the same time. Thus using nearly double the power while on the internet (surf) and voice (talk) at the same time.
With that being said, both AT&T and Verizon have rolled out HD Voice. HD Voice is a fancy way of saying that the voice part of the call is being done over the internet. AKA VoIP. This means that both AT&T and Verizon, the phone needs to be on the LTE network to work. Or be with in range of WiFi with WiFi calling enabled on the phone.
A friend of mine, who has an AT&T iPhone was able to be on a VoIP call at home on his WiFi and leave the house and it'll transfer to AT&T VoLTE seamlessly. However, he could not move from VoLTE back to the WiFi until he was done with the call. Then a fix came out a month or so later. So now he can.
KOT