The content generated might be unsafe for children or offensive, as the AI model has been trained with internet text data. While OpenAI has done work to mitigate the risks of offensive content generation, and mostly generates content that is fine, it is not a solved problem and occasionally offensive content will be produced. I recommend that you do not let your children interact alone with this tool.
Privacywise, I'm not storing any of your personal information. I don't care about that stuff. Just have fun.