If you allow the candidate to do what they'd do at their desk you can ask more than trivial questions. Certainly, if they have to ping google to figure out how to write fizzbuzz, that's a red flag, but it depends what the person spec is.

How many positions would you estimate (I'm an academic so have no experience) where encyclopedic and gap-less knowledge is required to accomplish a task? I envision a technical interview as an opportunity to make sure the candidate can actually do the task you need them to - the screening of non-programmers should be done beforehand.