We have the same issue. Contact Form works normally, if the user is logged in. As soon as the user is unknown to WordPress the form does not work.
We do not have any caching plugin in use so the idea from @pca2014 does not solve it.
Currently these plugins are active, but even when deactivating them, the problem persists:
– Flamingo
– Page Builder by SiteOrigin
– SiteOrigin Widgets Bundle
– TablePress
– Black Studio TinyMCE Widget
– Contact Form 7
