We present Paired by the Teacher (PbT), a two-stage teacher-student pipeline that synthesizes accurate input-output pairs without human labels or parallel data. In many low-resource natural language generation (NLG) scenarios, practitioners may have only raw outputs, like highlights, recaps, or questions, or only raw inputs, such as articles, dialogues, or paragraphs, but seldom both. This mismatch forces small models to learn from very few examples or rely on costly, broad-scope synthetic examples produced by large LLMs. PbT addresses this by asking a teacher LLM to compress each unpaired example into a concise intermediate representation (IR), and training a student to reconstruct inputs from IRs. This enables outputs to be paired with student-generated inputs, yielding high-quality synthetic data. We evaluate PbT on five benchmarks-document summarization (XSum, CNNDM), dialogue summarization (SAMSum, DialogSum), and question generation (SQuAD)-as well as an unpaired setting on SwitchBoard (paired with DialogSum summaries). An 8B student trained only on PbT data outperforms models trained on 70 B teacher-generated corpora and other unsupervised baselines, coming within 1.2 ROUGE-L of human-annotated pairs and closing 82% of the oracle gap at one-third the annotation cost of direct synthesis. Human evaluation on SwitchBoard further confirms that only PbT produces concise, faithful summaries aligned with the target style, highlighting its advantage of generating in-domain sources that avoid the mismatch, limiting direct synthesis.