You can move any existing template by changing its visibility scope.
Manage global templates:
You can navigate to the global templates page in two ways:
Click the cog icon in the Jira top bar, and click Manage Apps. Then, from the sidebar, find Manage Responses under CANNED RESPONSES FOR JIRA.
In the issue view, click the cog icon next to the Canned Responses button. Then, on the My Canned Responses page, click Global templates.
Here you can edit, clone, and delete your templates, as described in the “Canned Responses settings page” section above.
Organize templates into folders:
You can group templates into a hierarchical structure using a double colon in the template name. By adding double colons, you can separate the folder from the templates that belong to it. This is especially great for organizing templates if you have a large number of them. For example, to put the "Deleting account" template in the "Instructions" folder, you can name the template Instructions::Deleting account and it'll be nested accordingly.
When browsing through the templates inside the field where Canned Responses is available, you will see the folders with a little arrow next to them, indicating you can open them to select a template. Check out the example below:
Converting existing comments to new templates
If you think a comment may be useful in the future, you can create a template from it. After writing your message, click on the “+” icon next to the CR dropdown and save the content for future use.