The new wave of Large Language Models (LLM) has offered an efficient tool to curate sizeable conversational datasets. So far studies have mainly focused on task-oriented or generic open-domain dialogs, and have not fully explored the ability of LLMs in following complicated prompts. In this work, we focus on personalization, and employ LLMs to curate a dataset which is difficult and costly to crowd-source: PersonalityChat is a synthetic conversational dataset based upon the popular PersonaChat dataset, but conditioned on both personas and (Big-5) personality traits. Evaluating models fine-tuned on this dataset, we show that the personality trait labels can be used for trait-based personalization of generative dialogue models. We also perform a head-to-head comparison between PersonalityChat and PersonaChat, and show that training on the distilled dataset results in more fluent and coherent dialog agents in the small-model regime.