We will provide an overview of available Search notifications.
Templates for each language can be viewed and edited from Admin > System > Localisation > Edit templates > Search

Notification not sending?
Remember: The type of notification received by users, e.g. email, in-system, none, etc., will depend on their preferences as outlined here.
If a notification is not evident, check that the recipient's preferences are set to receive this as they would expect first.
Both notifications relate to 'User suggestions' being enabled by an application administrator on the admin side of Search:

Then, in the Search recommendations tab > User requests the two checkboxes correspond to the possible notifications:

search.suggestion_request_created
This notification will fire when a new search suggestion is created by a user. All Search application administrators will be notified, as long as the 'notify admin' option is selected as shown above.
search.suggestion_request_implemented
This notification will fire when a search suggestion made by a user has been implemented as a phrase in a search recommendation. The recipient is the user who created the suggestion.
(The phrase must exactly match the user suggestion for the system to be able to recognise this notification)