Here you can manage your notifications by feature.
In-app notifications are always on but you can choose to set up to also receive email notifications.
Inbox & Listening
You can choose to select all or individual profiles
Inbox item assigned - Sent each time a user assigns you an inbox item to review
Inbox Assistant rule - Sent each time an Inbox Assistant rule assigns you an inbox item to review
Listening item assigned - Sent each time a user assigns you a listening item to review
Publishing
These are the publishing notification available:
Publishing parent post assigned - Sent each time a user asks you to review a publishing parent post
Publishing parent post commented - Sent each time a user comments on a publishing parent post you've worked on
Publishing parent post rejected - Sent each time a user rejects a publishing parent post you've worked on
Publishing parent post approved - Sent each time a user approves a publishing parent post you've worked on
Publishing parent post updated - Sent each time a user edits a publishing parent post you've worked on
Publishing post failed - Sent each time one of your publishing items was not published
You can choose to select all or individual profiles for the Publishing post failed notifications.
Note: You only see the notifications you are allowed to receive. ie: a moderator on all the social profiles > you will be able to manage the Inbox notifications but not the Publishing ones
Check out this article for more information on roles and their permissions.