You have the option to be notified through e-mail or real-time (push) notification whenever you receive a starlog message. Some message, e.g. the conclusion of an exploration mission are marked as low priority, and you can choose to omit these for the real-time notifications.

Before push notifications can be sent, you must permit them, separately for each browser. You will be prompted for permissions when you enable the notifications, but will also see a warning on the starlog when they are enabled without permissions, and can trigger the request again there.

On most systems, for push notifications to work, the browser must be active. Some browser's have an option to allow them to run in the background even when they are closed to receive such messages.