The Directories app can allow a user to edit a record by clicking a button. To restrict which users should have the right to edit a record via the Directories app, the Owner Field option is included, and found at Directories > (selected directory) > Options tab > Parameters - Record Display.
This option should be populated with a Salesforce field name (including __c at the end for custom fields) within the object being searched in the directory that represents a Contact Id. Most commonly this field will be a lookup field to the Contact object. For example, if there is a field from the Case object with name "Case_Contact__c" that should be used to evaluate if the logged-in user should have the right to edit a record, the value to enter into the Owner Field is "Case_Contact__c".
With this option set by the administrator, when a visitor views a Directories app record view, the Directories app checks the value associated with the Owner Field for the record viewed, and evaluates if it's the same as the Contact Id of the logged-in user. If it is the same, the user will see an edit button and be given the right to edit the record. If not, the visitor will not see the edit button.