You are not required to use gmail or yahoo in order to send mail with Jira Server. There are detailed steps to follow in regards to how you can configure Jira to use an SMTP mail server for sending mail in Configuring an SMTP mail server to send notifications.
Currently, I don't believe that Jira Cloud can send notifications on a separate domain, but this is something you should be able to achieve if you're using Jira Server.
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