Natural Language Processing tasks that aim to infer an author's private states, e.g., emotions and opinions, from their written text, typically rely on datasets annotated by third-party annotators. However, the assumption that third-party annotators can accurately capture authors' private states remains largely unexamined. In this study, we present human subjects experiments on emotion recognition tasks that directly compare third-party annotations with first-party (author-provided) emotion labels. Our findings reveal significant limitations in third-party annotations-whether provided by human annotators or large language models (LLMs)-in faithfully representing authors' private states. However, LLMs outperform human annotators nearly across the board. We further explore methods to improve third-party annotation quality. We find that demographic similarity between first-party authors and third-party human annotators enhances annotation performance. While incorporating first-party demographic information into prompts leads to a marginal but statistically significant improvement in LLMs' performance. We introduce a framework for evaluating the limitations of third-party annotations and call for refined annotation practices to accurately represent and model authors' private states.