If you want anyone (even users without a JIRA account) to be able to view a specific Agile board, you can do so by setting up a specific permission scheme on the project.
In this example, we will be creating a new project and associate it with a new permission scheme.
If you already have a project created you can jump to Setting up permissions (assuming you either have an exclusive permission scheme or want to grant public access to all projects associated with that scheme).
All steps presented below require an account with JIRA Administrators or JIRA System Administrators global permission.
Creating an Agile-enabled project
On the Agile drop-down menu (in the header bar of JIRA), click Manage Boards;
Near the top of the page, click Create board;
Select any of the board types;
Choose the board and project's name;
Select any workflow type;
Creating a permission scheme
In order not to affect other projects in our instance, we'll create a brand new permission scheme.
Go to Administration > Issues > Permission Schemes;
Near the bottom of the page, click Add permission scheme;
Give the new scheme a name;
Setting up permissions
When clicking Permissions on the right of the newly-created scheme, you'll see there isn't any grants. You can now setup all the permissions as you wish. For this case, the only permission that matters is the Browse Projects one and, for you to maintain full-access to the project, the Administer Project has to be granted to a group you pertain.
The Browse Projects permission has to include the Anyone group. To do so, follow the below steps while on the Permissions page of your scheme.
On the right of the Browse Projects permission, click Add;
Select the Group permission;
Make sure Anyone is selected on the drop-down menu to the right;
Associating the permission scheme with a project
Now, we have to tell our project to use the newly-created scheme, otherwise it will keep using the default one.
Go to the Project Administration page;
Click Permissions on the sidebar;
Near the top of the page, on the Actions drop-down menu, click Use a different scheme;
Select the scheme you created;
Your project can now be accessed by anyone who can connect to your JIRA instance and there's no need to login. Users will be redirected to the Login page, however you can use the drop-down menus on the top bar to navigate to the project or inform the users the URL for direct access to the board.