How can I create one Board with two users/user groups that don't see each others' tickets?

[Nic Brough -Adaptavist- · Accepted solution]

A board will not display or count issues that the viewer cannot see.  So you just create a board, and let JIRA handle it based on who is logged in and looking at it.

Or do you mean that you want the board to display issues that a user cannot normally see?  That would be a security issue, and hence never going to happen - you'll need to allow your users to see the issues you want them to work with.