Co-design is a collaborative and participatory design (PD) method frequently used in face-to-face settings. However, in-person studies can be difficult or even impossible to conduct in certain situations due to different constraints. Thus, an increasing line of work has been relying on methods that allow researchers to conduct PD studies remotely. In this paper, we cover findings of a case study with 25 participants being part of asynchronous and synchronous remote co-design workshops. It was found that there are important differences between the two modalities regarding participation and engagement, activity design, as well as issues with privacy and confidentiality. We also provide methodological implications and actionable steps to conduct remote co-design studies both synchronously and asynchronously more effectively.