Vision-Language Models (VLMs) combine visual and textual understanding, rendering them well-suited for diverse tasks like generating image captions and answering visual questions across various domains. However, these capabilities are built upon training on large amount of uncurated data crawled from the web. The latter may include sensitive information that VLMs could memorize and leak, raising significant privacy concerns. In this paper, we assess whether these vulnerabilities exist, focusing on identity leakage. Our study leads to three key findings: (i) VLMs leak identity information, even when the vision-language alignment and the fine-tuning use anonymized data; (ii) context has little influence on identity leakage; (iii) simple, widely used anonymization techniques, like blurring, are not sufficient to address the problem. These findings underscore the urgent need for robust privacy protection strategies when deploying VLMs. Ethical awareness and responsible development practices are essential to mitigate these risks.