Right, unless you interpret the second half of the sentence as clarifying or defining the first. Which I do. If may not be your preferred definition of "private", but it is a definition of private.

To paraphrase it another way, it says "what you do on your browser is hidden, specifically it is hidden from other people who would try to find out using this device"

Further, given the overall ambiguity of English, and given the goal is something casual, not legalese, the way it is phrased is reasonable.

Actually, that raises an interesting point, how is this phrase rendered by chrome for other languages, and do any of them phrase it in a less ambiguous way?