Assuming users have "browse project" permission for the included projects, You must share the underlying filter of the board, which can be done at any time from the filter config/issue navigator view.
You would need the Browse Project permission of all the projects associated to the Scrum Board. Also, you may want to check the filter query in Board > Configuration > Filter tab.
More information on GreenHopper permission which relates to JIRA can be seen in the following https://confluence.atlassian.com/display/AGILE/JIRA+Agile+-+JIRA+Configuration#JIRAAgile-JIRAConfiguration-Permissions
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