How can I see a report of which participants have registered for a discussion board?

Exporting the user list will export all the user details including each user's registration status.  Here is how you do that.

Hover over the Users tab on your project or activity page and select Add/Edit Users. 

Click the Export button above the Users list.

An Excel file will download to your computer with a list of all your users.  Column I will note each participant's registration status.