The implications of using APIs from blackbox models versus local models in terms of privacy and liability are significant. When using APIs from blackbox models, it is safer to consider private data that has never been shared online to prevent any potential overlaps with pre-training data. This ensures better protection of user privacy and reduces liability risks associated with unintentional exposure of sensitive information. On the other hand, when using APIs from local models where users have full control over model weights and architectures, they can take additional precautions to ensure no overlap between private and pre-training data for enhanced security.