What Badges can see, and what it keeps
Badge records, your programme and your events. Badges asks OSM for read-only access to badges, and to your programme and events so a report can say which upcoming meeting or camp covers a requirement, and who's going. A badge record includes each young person's name and Six or Patrol, which is what the report prints.
Contacts, only for parent emails. If you connect parent emails, Badges also reads primary contacts' names and email addresses, when you send. OSM's permission for that covers medical details too; Badges doesn't read them.
Nothing is stored. Each report is built from OSM when you open it and thrown away afterwards. What Badges keeps is your OSM connection (encrypted, so it can refresh without you signing in to OSM every time), your group's report name and logo, and for parent emails a record of each one sent (with the address partly hidden) and of families who asked to stop. Badges' privacy notice has the detail.
Who can use it. Anybody your group's owner has given access to Badges on the Members page. A leader in Badges can change the report name and logo; anybody with access can print reports for the sections they have in OSM.