tl;dr iPhone 6s either model, will work on both carriers with simultaneous voice and data, iPhone 7, only VZW model will work on both carriers, both excellent world phones.

Iphone 6, 6s, and SE are the same on VZW and AT&T. Both can do simultaneous Voice and Data, due to everything going over LTE. 5S and below on verizon can not do simultaneous voice and data due to doing calling over CDMA based 3G. iPhone 6 is a little limited when it comes to being a global phone. The 6S is excellent as the only difference between the AT&T and the VZW model is one LTE band which the only carrier to use that one band in the world, is AT&T. But the 6S, either model will work with almost any carrier in the world with full compatibility sans a few in Japan. iPhone 7 on Verizon has a radio that has full compatibility on almost every network in the world sans a few in Japan. AT&T iPhone 7 will not work on any cdma (important in US with VZW and sprint and some asian countries) or tdscdma networks (important in china) so that may hinder you if you want to ever go there. The unlocked iPhone 7 is the same model as the VZW iphone 7 and will work with A&T and has shown to be a touch faster, and shows that it can handle carrier aggregation with both carriers. Just some technical input. willmyphonework.net is also a cool tool.