I was having the same issue of jetpack contact form not sending emails. Feedbacks were coming through but not emails.
Under settings, discussion settings, before a comment appears.
I unchecked, Comment author must have a previously approved comment. I then saved the change. I tested the comment form and it worked.
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The Mexykan on "[Plugin: Jetpack by WordPress.com] Jetpack Contact Form Not Sending Emails"
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