Does this work with or is it a replacement for authcache?

you are right

there is no sense to use both.
ESI is more complex way, because you have to install varnish as reverse proxy, config VCL.
But Varnish is better for highload.

Authcache easy to config and debug side-effects.

@SocialNicheGuru, did you get an answer for

Can you tell me if this works with authcache or is it a replacement for it?
I think esi replaces the personalized aspects of a page that autcache can then cache for an authenticated user.

I have the same question...

http://drupal.org/node/1939742#comment-7250734
yes it is

Do you mean that authcache works with ESI module? Our requirement is configure varnish for authenticated user and then use ESI, any recommended steps?

no. ESI + varnish is other way to have the same functionality that authcache provide.

Our requirement is configure varnish for authenticated user and then use ESI, any recommended steps?
ESI module does not provide varnish config. If you use Drupal 6 you can check easy authcache 2.x-dev module that can generate varnish config.

Check dependences. I think no

You don't need the varnish module…but if you're running a high-traffic site, the varnish module can help by sending 'clear cache' instructions to varnish when content changes. The expire module can also help, by improving the granularity of the 'clear cache' instructions, for example, so that it clears the cache of just a specific node page.