Frequent public uproar over forms of data science that rely on information about people demonstrates the challenges of defining and demonstrating trustworthy digital data research practices. This paper reviews problems of trustworthiness in what we term pervasive data research: scholarship that relies on the rich information generated about people through digital interaction. We highlight the entwined problems of participant unawareness of such research and the relationship of pervasive data research to corporate datafication and surveillance. We suggest a way forward by drawing from the history of a different methodological approach in which researchers have struggled with trustworthy practice: ethnography. To grapple with the colonial legacy of their methods, ethnographers have developed analytic lenses and researcher practices that foreground relations of awareness and power. These lenses are inspiring but also challenging for pervasive data research, given the flattening of contexts inherent in digital data collection. We propose ways that pervasive data researchers can incorporate reflection on awareness and power within their research to support the development of trustworthy data science.