Does the logbook app work offline?
Yes. In the store listing's words, it “works completely offline, so you can log a watch mid-ocean with no signal”. Signing on, starting and ending a voyage, logging a watch, adding past service and editing voyages all save straight to your phone without waiting for a network.
What else works offline?
Your totals and Career screen, which are worked out from what's on the phone; your certificates and their phone reminders, which are scheduled on the device; and switching the app's language, which happens instantly with no download.
What needs a connection?
In practice: signing in, the backup itself, sharing a PDF or link by email or message, the expiry email that backs up the phone reminders, and buying Bridgedeck Pro. Your public sea-service profile, if you turn it on, shows your backed-up figures, so it updates after your logbook syncs.