Connected vehicles enable a wide range of use cases, often facilitated by smartphone apps and involving extensive processing of driving-related data. Since information about actual driving behavior or even daily routines can be derived from this data, the question of privacy arises. We explore the impact of privacy assurances on driving data sharing concerns. Specifically, we consider two data-intensive cases: usage-based insurance and traffic hazard warning apps. We conducted two experimental comparisons to investigate whether and how privacy-related perceptions about vehicle data sharing can be altered by different types of text-based privacy assurances on fictional app store pages. Our results are largely inconclusive, and we did not find clear evidence that text-based privacy guarantees can significantly alter privacy concerns and download intentions. Our results suggest that general and threat-specific privacy assurance statements likely yield no or only negligible benefits for providers of connected vehicle apps regarding user perceptions.