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: https://sendbird.com/docs/chat/v3/javascript/guides/push-notifications#1-push-notifications
iOS: https://sendbird.com/docs/chat/v3/ios/guides/push-notifications#1-push-notifications
Android: https://sendbird.com/docs/chat/v3/android/guides/push-notifications#1-push-notifications
Flutter: https://sendbird.com/docs/chat/v3/flutter/guides/push-notifications#1-push-notifications