What's the "value" of finding and eliminating a total duplicate site hosted under another domain?

[Jake]

Both sites are your properties?

Search engines reward sites for having valuable original content, there is less value in what is identified as duplicate material. If you have a specific site you want to get the ranking fom te material you can use the cannonical tag on the secondary site to point to the official source of the material, giving that site the ranking credit. While mainly for indicating the master copy within a site cannonical can also be used to point to another source site
http://www.seomoz.org/blog/cross-domain-...ard-friday
http://www.seomoz.org/blog/complete-guid...nd-why-not
http://www.seomoz.org/q/similar-pages-no...parameters

A measurable site traffic factor is the search engine ranking, the #1 result in search results has been find in some studies to get about 35% of all clicks, moving up 2 positions in the page 1 search results gets approximately double the clicks. After you have compiled a list of valuable real world searh phrases, perhaps using the Google Keyword Tool https://adwords.google.com/select/KeywordToolExternal
You can periodically use rank checking tools to automatically check your site ranking in sarch results for the entire keyword list. (That same keyword list should be consulted when creating content for the site)

There are tools that will report your search result position for various search keywords:
Use either the free rankchecker for firefox http://www.link-assistant.com/rank-tracker/ to generate a search ranking report for the list of keywords, the latter has some other analysis functions. (The paid version of SEO Powersuite probably makes sense in your business, it allows saving results for month by month comparison)

Remember to try the same search phrases, with: ,your city appended to the ends for local ranking.