The development of Natural Language Generation models has led to the creation of powerful Artificial Intelligence-assisted writing tools. These tools are capable of predicting users' needs and actively providing suggestions as they write. In this work, we conduct a comparative user-study between such tools from an information retrieval lens: pull and push. Specifically, we investigate the user demand of AI-assisted writing, the impact of the two paradigms on quality, ownership of the writing product, and efficiency and enjoyment of the writing process. We also seek to understand the impact of bias of AI-assisted writing. Our findings show that users welcome seamless assistance of AI in their writing. Furthermore, AI helped users to diversify the ideas in their writing while keeping it clear and concise more quickly. Users also enjoyed the collaboration with AI-assisted writing tools and did not feel a lack of ownership. Finally, although participants did not experience bias in our experiments, they still expressed explicit and clear concerns that should be addressed in future AI-assisted writing tools.