I think if the other party is wearing a Ring of Power themselves, of any kind, that would change the dynamic a lot, yes. The Nazgûl could see Frodo just fine when he was wearing it--and just as significantly, he could see them. If he'd put it on in Lothlórien, I wonder if Galadriel could have seen him. The Nazgûl might be a special case, though, since they're undead creatures who've been so utterly transformed by their Rings that they exist in that otherworldly zone all the time. That didn't happen to either elves or dwarves in the same way.

For the purposes of this question, let's assume the other partner is Ringless.