The use of Natural Language Processing (NLP) in high-stakes AI-based applications has increased significantly in recent years, especially since the emergence of Large Language Models (LLMs). However, despite their strong performance, LLMs introduce important legal/ethical concerns, particularly regarding privacy, data protection, and transparency. Due to these concerns, this work explores the use of Named-Entity Recognition (NER) to facilitate the privacy-preserving training (or adaptation) of LLMs. We propose a framework that uses NER technologies to anonymize sensitive information in text data, such as personal identities or geographic locations. An evaluation of the proposed privacy-preserving learning framework was conducted to measure its impact on user privacy and system performance in a particular high-stakes and sensitive setup: AI-based resume scoring for recruitment processes. The study involved two language models (BERT and RoBERTa) and six anonymization algorithms (based on Presidio, FLAIR, BERT, and different versions of GPT) applied to a database of 24,000 candidate profiles. The findings indicate that the proposed privacy preservation techniques effectively maintain system performance while playing a critical role in safeguarding candidate confidentiality, thus promoting trust in the experimented scenario. On top of the proposed privacy-preserving approach, we also experiment applying an existing approach that reduces the gender bias in LLMs, thus finally obtaining our proposed Privacy- and Bias-aware LLMs (PBa-LLMs). Note that the proposed PBa-LLMs have been evaluated in a particular setup (resume scoring), but are generally applicable to any other LLM-based AI application.