Many sensitive domains -- such as the clinical domain -- lack widely available datasets due to privacy risks. The increasing generative capabilities of large language models (LLMs) have made synthetic datasets a viable path forward. In this study, we domain-adapt LLMs to the clinical domain and generate synthetic clinical texts that are machine-annotated with tags for personally identifiable information using capable encoder-based NER models. The synthetic corpora are then used to train synthetic NER models. The results show that training NER models using synthetic corpora incurs only a small drop in predictive performance. The limits of this process are investigated in a systematic ablation study -- using both Swedish and Spanish data. Our analysis shows that smaller datasets can be sufficient for domain-adapting LLMs for data synthesis. Instead, the effectiveness of this process is almost entirely contingent on the performance of the machine-annotating NER models trained using the original data.