In one of her first Karen appearances Huntress put it down to the fact no one would suspect Power Girl was sitting in a restaurant. Since she didn't have many solo appearances to back up her ID she never fell into the usual plothole of having people who knew Karen also being connected to PG. And i think her love interest, Andrew Vinson, had been in on establishing her secret ID. So Karen was simply one more buxom blonde on the street to anyone she didn't tell differently.
Clearly, from the sparce answers to this question found in some incredibly lame continuity, the creative team for the new book won't have to work hard to cover new ground, will they? Despite being around 30 years, the character obviously is an open book.