My guess is that what the OP was thinking of is whether Verizon and ATT could use each other's towers. For roaming.
So even if ATT doesn't have coverage in North Dakota, and Verizon blanketed it with LTE, ATT customers might be able to use their phones in ND.
I suspect that it will work like it does now with compatible technologies. You won't be able to roam on the other company's towers if your company thinks you should have coverage in the area you are in. If, for instance, ATT has Edge coverage in an area, and Verizon has LTE, you probably won't be able to use the LTE. Even if you can't pick up the Edge service. But in a whole area that ATT isn't covered, they may be able to negotiate roaming service.
Roaming is expensive.
I can remember a couple years ago in a hotel in Miami. My sister has TMobile. I have ATT. On the ocean side of the hotel, I was lucky to get text messages through. She had great coverage. Other side of the hotel, I had decent coverage. Since ATT had coverage in the area, the phone wouldn't roam.