I don't quite understand what you mean by "having a say in each other's rings." We both got what we wanted but there wasn't really anything to argue over because we both got modest, regular looking rings. Much later, when we had more money, I got an upgrade.
Is the problem the look of one of the rings in some way or the price or does one want matching rings and the other doesn't or...?
I think I'm just looking for an idea of how to set up the process of both of us getting input.
I never compromised on original e-ring or subsequent upgrades. We
came up with a budget together and shopped within my budget. I guess my original wedding band was a bit of a compromise because we were poor and so I had limited budget. I got a plain 2mm platinum band that I wear mostly as a spacer now. My husband had 2 wedding bands, first a platinum band, then a palladium band, but he lost both because he takes it on and off for his Job and gym. He has not worn a wedding band now for about 5 years, and I’m fine with that. That is our compromise.