Does Sendbird support Push Notifications when the user is offline/logged out?

Yes. A notification can be sent to the user’s device and be accessed by the user even if the user is logged out of the app. The notifications preferences can be set by the customer (reference)

You can find more information here:
Javascript: Push notifications | Chat JavaScript SDK | Sendbird Docs
iOS: Push notifications | Chat iOS SDK | Sendbird Docs
Android: Push notifications | Chat Android SDK | Sendbird Docs
Flutter: Push notifications | Chat Flutter SDK | Sendbird Docs