A board is available to all users who can view the filter on which the board is based. (Note that they will also need 'Browse project' permission for the project(s) whose issues are shown on the board.)
To configure a board:
Then you will have to edit the board filter settings;
This one should be shown when configuring the board.
Filter Query: project in (EXAMPLE) ORDER BY Rank ASC
So, in this board filter you should make it so that only certain groups can see it. You can for example share it with certain groups or projects.
If you want external contacts to have access then they will need permissions to a Jira project where your issues are. Those issues are then pulled into a board.
Have you seen these instructions?
Please make sure you test this first. Let me know how you get on. Happy to walk you through this if easier.
If you want them to have access to 1 board or project then you really must do what's suggested here: https://confluence.atlassian.com/jirakb/how-to-restrict-project-access-to-different-isolated-user-groups-290750862.html?_ga=2.44787044.1901907075.1509353518-204997263.1500891987