My husband and I don't wear rings but we were not in agreement. I could have gone either way originally but I didn't get to weigh in on the matter ultimately, and that is the only reason it was an issue. (long story)
Whatever you BOTH decide to do is just fine in my opinion!
I will add that we are often asked why we don't have rings and many assume we are not married.
We go to public events fairly often and many times I have been asked out by men with my husband standing right next to me
. I can't help but laugh a little when it happens, its not the guys fault, I'm not wearing a ring! Also, my husbands close friends are very disapproving of our lack of rings.
But again, if you both are ok without, I think that's fine, just an fyi on our experience with other folks.