Does the Auto-Cache Engine run for logged-in users?
The Auto-Cache Engine (WordPress Dashboard → Comet Cache → Plugin Options → Auto-Cache Engine) is designed to pre-cache your site, that is to generate cache files ahead of time so that visitors don't have to wait for the cache file to be generated if they are the first to visit a page.
When Logged-In User Caching is enabled (WordPress Dashboard → Comet Cache → Plugin Options → Logged-In Users), Comet Cache will maintain a separate cache for each user that logs into your site. However, the Auto-Cache Engine will not generate cache files for logged-in users because doing so would require that the Auto-Cache Engine log in as each user on your site, and then visit each page on your site to generate a cache file for that user. That is not practical because it does not scale.
A site that allows visitors to login generally has pages that are dynamic to the user logging in, so anything that generates a cache file ahead of time wouldn't be a good idea anyway. The Auto-Cache Engine will only cache pages that non-logged-in visitors can see, as per your XML Sitemap and/or your List of URLs to Auto-Cache.
If you are using Logged-In User Caching, Comet Cache will generate a user-specific cache file at the time the user visits each page for the first time and it will then do its best to make sure that the user-specific-cache remains up-to-date.