User privacy is a requirement for wireless vehicular communications, and a number of privacy protection strategies have already been developed and standardized. In particular, methods relying on the use of temporary pseudonyms and silent periods have proved their ability to confuse attackers who would attempt to track vehicles. In addition to their ability to protect privacy, it is important to ensure that these privacy strategies do not hinder the safety applications which rely on vehicular communications. This paper addresses this concern and presents an experimental analysis of the impact of privacy strategies on Intersection Collision Avoidance (ICA) systems. We simulate traffic scenarios at a road intersection and compare the ability of a collision avoidance system to avoid collisions for different pseudonym change schemes. The privacy level is analyzed, as well as the influence of the duration of the silent period on the safety performance of the ICA system. The results highlight the need to jointly design safety applications and privacy strategies.