Activate the JIRA Paste Plugin
Steps
Installing the JIRA CKEditor Integration extension is not enough to turn pasted JIRA URLs into JIRA Macros: the plugin it provides has no toolbar button, so it is only loaded once its name is declared in the CKEditor configuration. It also needs at least one configured JIRA instance.
- Log in as a user having administration rights on the wiki.
- Go to the administration and open "Editing", then "WYSIWYG Editor".
- Select the "CKEditor" tab and find the "Advanced configuration" field.
- Write config.extraPlugins = 'xwiki-jira-paste'; in that field.

- Click "Save".
- Edit a page with the WYSIWYG editor and paste the URL of an issue of a configured JIRA instance. It is replaced by a JIRA Macro.

FAQ
Is this needed to display issues with the JIRA Macros?
No. The JIRA Macros work as soon as they are inserted in a page. Only the conversion of a pasted URL depends on this configuration.
The "Advanced configuration" field already contains something. What do I do?
Keep what is there and declare the plugin alongside the others in the same comma-separated list, rather than replacing the content of the field.
Pasting a JIRA URL still inserts a plain link. What is missing?
Either the plugin is not declared, or no JIRA instance is configured, or the pasted URL does not start with the URL of a configured instance, or the paste was done without formatting, which deliberately never converts.
Where does the name "xwiki-jira-paste" come from?
It is the identifier of the CKEditor plugin the extension registers, and it is the only value this extension requires in "Advanced configuration".