Service robots often perform their main functions in public settings, interacting with more than one person at a time. How these robots should handle the affairs of individual users while also behaving appropriately when others are present is an open question. One option is to design for flexible agent embodiment: letting agents take control of different robots as people move between contexts. Through structured User Enactments, we explored how agents embodied within a single robot might interact with multiple people. Participants interacted with a robot embodied by a singular service agent, agents that re-embody in different robots and devices, and agents that co-embody within the same robot. Findings reveal key insights about the promise of re-embodiment and co-embodiment as design paradigms as well as what people value during interactions with service robots that use personalization.