Manifyst stores what a signup form collects: first and last name, any email address or phone number provided, notes, the answers to custom fields, and the timestamps around the signup.
It plugs into WordPress's own privacy tools, so requests are handled where you already handle them, under Tools → Export Personal Data and Tools → Erase Personal Data.
Export returns every signup matching the requester's email address, under the group "Manifyst Event Signups", alongside whatever other plugins contribute.
Erasure anonymizes rather than deletes. The personal details are stripped, but the registration itself is kept so that slot capacity and the audit history stay accurate — otherwise erasing one person would silently change how full a shift appears to have been. WordPress reports this back on the erasure screen.
Manifyst also contributes suggested text to Settings → Privacy → Privacy Policy Guide. Review it and paste what applies into your policy; it describes what the plugin collects, but only you can say why you collect it and how long you keep it.
Note that erasure matches on email address. A signup made with a typo'd address won't be found by a request from the correct one.