I am fine with white board type situations, don't worry about writing code without IDEs etc (in my very first job vi editor was the only thing we had to write code so I still have instinctive memory to be resourceful without an IDE but I love IntelliJ for most things). Happy to discuss/draw out ideas/solutions on whiteboards.
But find it very seriously annoying talking or being expected to talk while I am writing code !
I think the internet-access requirement depends on the problem. If I am expected to solve a problem by first-principles on a core CS topic then I guess internet is of no use ? Online Java Docs for e.g. may not serve a purpose here.
If the problem requires using many different tools/APIs for the solution then I guess you are right, internet access adds value.