Both systems have a "footer" bar which is shared across all pages, and can be modified to remove version information and add your own (it's coding, but it's quite simple). You must retain the powered by atlassian link in there, but you can do anything you want with the rest.
As far as I can remember, the footer is the only place versions are routinely exposed to end-users.
See https://confluence.atlassian.com/jira061/jira-administrators-faq/usage-faq/modifying-the-jira-footer and https://confluence.atlassian.com/display/CONFKB/How+to+edit+the+footer+in+Confluence
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