We would need answers to the following questions to determine the complexity of this task, or in general to actually implement it:

• Name of the log (first selector on Special:Log)
  • I would suggest using a generic name here, and then maybe having subtypes (cfr. "type of deletion" input on Special:Log/delete) for registered vs unregistered vs changed registration; I know there would only be one subtype for now, but this might change in the future. In this case, we would also need the label of this selector, and the names of its options.
• Content of the header below the page title (for the deletion log, this is "Below is a list of the most recent deletions."). Remember that this should describe all subtypes.
• Who should be able to view this log? Should be public unless there are good reasons against it.

If someone switches from public to private, the log associated with their registration should be removed; is this possible?

Logs are never removed. The closest thing to this would be to save the log entry ID somewhere, and then suppress it if the participant becomes private. Again, this is hacky, expensive, perhaps surprising UX, and may not work in all cases.

Note that logging is per-wiki and not global in terms of what users can see in Special:Log

And to clarify, logging does not happen on the wiki where the event page is, but on the wiki where the participant registered. So the log entries for a single event may be scattered across many wikis.