Modern Natural Language Generation (NLG) models come with massive computational and storage requirements. In this work, we study the potential of compressing them, which is crucial for real-world applications serving millions of users. We focus on Knowledge Distillation (KD) techniques, in which a small student model learns to imitate a large teacher model, allowing to transfer knowledge from the teacher to the student. In contrast to much of the previous work, our goal is to optimize the model for a specific NLG task and a specific dataset. Typically in real-world applications, in addition to labeled data there is abundant unlabeled task-specific data, which is crucial for attaining high compression rates via KD. In this work, we conduct a systematic study of task-specific KD techniques for various NLG tasks under realistic assumptions. We discuss the special characteristics of NLG distillation and particularly the exposure bias problem. Following, we derive a family of Pseudo-Target (PT) augmentation methods, substantially extending prior work on sequence-level KD. We propose the Joint-Teaching method, which applies word-level KD to multiple PTs generated by both the teacher and the student. Finally, we validate our findings in an extreme setup with no labeled examples using GPT-4 as the teacher. Our study provides practical model design observations and demonstrates the effectiveness of PT training for task-specific KD in NLG.