Understanding how exposure to news on social media impacts public discourse and exacerbates political polarization is a significant endeavor in both computer and social sciences. Unfortunately, progress in this area is hampered by limited access to data due to the closed nature of social media platforms. Consequently, prior studies have been constrained to considering only fragments of users' news exposure and reactions. To overcome this obstacle, we present an innovative measurement approach centered on donating personal data for scientific purposes, facilitated through a privacy-preserving tool that captures users' interactions with news on Facebook. This approach offers a nuanced perspective on users' news exposure and consumption, encompassing different types of news exposure: selective, incidental, algorithmic, and targeted, driven by the diverse underlying mechanisms governing news appearance on users' feeds. Our analysis of data from 472 participants based in the U.S. reveals several interesting findings. For instance, users are more prone to encountering misinformation because of their active selection of low-quality news sources rather than being exposed solely due to friends or platform algorithms. Furthermore, our study uncovers that users are open to engaging with news sources with opposite political ideology as long as these interactions are not visible to their immediate social circles. Overall, our study showcases the viability of data donation as a means to provide clarity to longstanding questions in this field, offering new perspectives on the intricate dynamics of social media news consumption and its effects.