Hi Javier,
In ESR, you can subscribe for some types of objects (an option "Manage Subscription" in an object's context menu). When done so, if such an object gets modified or deleted, you (and other subscribers) will receive a notification e-mail. This functionality is not available for literally any ESR object: it is only available for interface objects (external definitions, data types and their enhancements, message types and fault message types, service interfaces, etc.), but not for other types of objects (mapping objects, imported objects, modeling objects, etc.). Using this functionality, you will also not be able to subscribe for a particular change, but you subscribe for an action type (for example, for all changes to an object, for its deletion). You can read more about this functionality at Managing Subscriptions - Managing Services in the Enterprise Services Repository - SAP Library.
In Integration Directory, this functionality is not available and I'm not aware of any analogous functionality available there.
Regards,
Vadim