With the fast-developing growth of conversational agents (CAs), particularly social CAs, this study aims to investigate users' intentions to share personal information. Users' more profound and broad engagement with social CAs has raised concerns over data privacy, so drawing on Protection Motivation Theory (PMT), the current study evaluates the effect of perceived severity threat and perceived data security vulnerability as the threat appraisals and self-efficacy in information security (SEIS), response efficacy, and response cost as the factors associated with coping appraisals on shaping disclosure behaviors with social ACs. The methodology includes a survey that evaluates the effect of social CA users' cognitive appraisals on their intention to self-disclosure. This study's findings help better understand the impact of security perception on users' behavior and provide recommendations for developers in creating more secure and transparent CA interactions.