If Verizon handed you a phone that only does CDMA, you still can't take it to ATT. And the other way around with a GSM only device. The upper end models can handle both systems these days, but the low end phones may not.
It may work better with the secondary carriers if they have contracts with Verizon and ATT/T-Mobile.
It may work better with the secondary carriers if they have contracts with Verizon and ATT/T-Mobile.