Using non public information obtained from a client in dealings with another client, or at all on the public side of the firm, would breach Chinese walls. Never mind internal controls, most jurisdictions have relevant legislation to prevent exactly this behaviour.
There is no non-public info in this case. It's same as if you were in a public hall with a lot of people, who talk only about stocks, and writing down how many, how often and when particular stocks are mentioned and in which context (buy, sell, various). Morally ambiguous thing is that hall belongs to Google and they're the only one up in the balcony, which is the only place you can hear all the people from.