Connected vehicles enable a wide range of use cases, often facilitated by smartphone apps and involving extensive processing of driving-related data. Since sensitive information about actual driving behavior or even daily routines can be derived from this data, the issue of privacy arises. We explore the impact of short privacy assurance statements on user perceptions by considering two data-intensive cases, usage-based insurance, and traffic hazard warnings. We conducted two experimental comparisons to investigate whether and how privacy-related perceptions about vehicle data sharing can be altered by different text-based privacy assurances on fictional app store pages. Our results are largely inconclusive, and we found no clear evidence that such short statements can significantly alter privacy concerns and increase download intentions. Our results suggest that some general and threat-specific privacy assurance statements might yield no or little benefits to connected vehicle app providers regarding user perceptions.