Define "productive." I define it as the marginal value added by a unit of labor input. Otherwise, productivity is just a feeling.
I define productivity as being to finish something. I believe the Internet helps me--a a programmer--produce more efficiently.
Some of that is social networks, because I'm hooked up to 400 or so programmers on twitter and often they can answer a simple question quicker than a Google search [despite the fact that I usually only go to Twitter to complain about my problem not to look for a solution].
Sites like StackOverflow--or even personal blogs--have saved me days or even weeks accomplishing "non-routine" tasks from consuming Twitter feeds to communicating with LDAP to securing a server with ColdFusion on it. I can accomplish such tasks more quickly by relying on the the knowledge of others; as opposed to having to discover it all myself.
[Repeat last paragraph for things like libraries and frameworks whose circulation would be much more limited without Internet access]
Of course sites like Hacker News and Facebook can suck up some of that productivity, but the alternative is much worse.
Your definition of "productive" requires subdefinitions for both "value" and "unit of labor". In a Taylorist environment natural definitions for these terms might be apparent and obvious, but for the average modern day office worker these terms are nearly impossible to define, much less compare across workers performing heterogeneous tasks in a complex environment where the final product or service is not simply the sum of its parts.
When confronted with exceedingly complex or impossible measurement problems, human intution has been and remains one of the best options we have. It may not be objectively accurate, especially at an individual level, but at least falling back to human intution when confronted with the unmeasurable allows correlations and changes over time to be observed.
The real question is how honest the source data is for this survey, not how precise or rigorous it is.
> One of the biggest benefits of the Internet, workers said, was the expanded network of contacts outside their company that they had built up.
“These are the new leads. These are the Glengarry leads. And to you they're gold, and you don't get them. Why? Because to give them to you would be throwing them away. They're for closers.”