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As autonomous systems begin to operate amongst humans, methods for safe interaction must be investigated. We consider an example of a small autonomous vehicle in a pedestrian zone that must safely maneuver around people in a free-form fashion. We investigate two key questions: How can we effectively integrate pedestrian intent estimation into our autonomous stack? Can we develop an online monitoring framework to give rigorous assurances on the safety of such human-robot interactions? We present a pedestrian intent estimation framework that can accurately predict future pedestrian trajectories given multiple possible goal locations. We integrate this into a reachability-based online monitoring and decision making scheme that formally assesses the safety of these interactions with nearly real-time performance (approximately 0.1s). These techniques are both tested in simulation and integrated on a test vehicle with a complete in-house autonomous stack, demonstrating safe interaction in real-world experiments.