An Exchange Administrator can check the configuration of a given mailbox using PowerShell:
Get-MailboxAutoReplyConfiguration -Identity <mailboxid>
There are several properties here, and they all work in different ways based on how it is enabled. What you want to watch out for:
AutoReplyState
: Enabled/Disabled
ExternalAudience
: None (default), Known (contacts in address book), All
Please pay attention to the Enabled state vs. Scheduled state. It is a little bit goofy.
Further reference on Technet: Set-MailboxAutoReplyConfiguration
ProTip: If you ever need to set a message for another user, draft the message in Word, and Save As HTML. You can import the HTML into a variable using Get-Content and then assign the InternalMessage/ExternalMessage using this variable. This method ensures the best experience for web/email, even though the HTML output from Word is gross.