What effect does BigPipe have if my traffic is mostly anonymous and I'm serving unpersonalized content behind varnish?
How does BigPipe have an effect if my traffic is mostly anonymous and behind varnish?
This really depends on the content of the page. Taking away any content depending on the logged-in user there might be no dynamic part left which could have any effect.
Additionally, why does the second request bypass Varnish?
As mentioned in the module description it is not possible the send the cache tag headers after the dynamically generated content has been streamed to Varnish. The Internal Page Cache can be controlled without sending headers because it is cached in the local database.