Effective robotic systems for long-horizon human-robot collaboration must adapt to a wide range of human partners, whose physical behavior, willingness to assist, and understanding of the robot's capabilities may change over time. This demands a tightly coupled communication loop that grants both agents the flexibility to propose, accept, or decline requests as they coordinate toward completing the task effectively. We apply a Mixed-Initiative dialog paradigm to Collaborative human-roBot teaming and propose MICoBot, a system that handles the common scenario where both agents, using natural language, take initiative in formulating, accepting, or rejecting proposals on who can best complete different steps of a task. To handle diverse, task-directed dialog, and find successful collaborative strategies that minimize human effort, MICoBot makes decisions at three levels: (1) a meta-planner considers human dialog to formulate and code a high-level collaboration strategy, (2) a planner optimally allocates the remaining steps to either agent based on the robot's capabilities (measured by a simulation-pretrained affordance model) and the human's estimated availability to help, and (3) an action executor decides the low-level actions to perform or words to say to the human. In physical robot trials with 18 unique human participants, MICoBot significantly improves task success and user experience over a pure LLM baseline and standard agent allocation models. See additional videos and materials at https://robin-lab.cs.utexas.edu/MicoBot/.