Well there would be an end to content duplication and copyrights scandals if this becomes known, the very reaosn why Google or any other search engine would never ever disclose this information.
Suppose a page surpasses the threshold of the level Search Engine has set for content duplication and it decides to take action, it is not known and definite which page SE would penalize; the one who copied the content or the one whose content was copied. Ofcourse this could be traced back as to when the first of those pages appeared on the web, but still it is not definite. So it is better to keep checking for copied content from your site as well.
As for your situation where two pages of the same site have similar content, it depends upon the situation. If it is only one and the only page, it might not do any harm. It is advisable in most of the cases to remove the duplicate page. If you cannot do that, use the canonical tag to indicate to the search engine which page it should consider while crawling. You can also noindex the page if you can afford to do that, though that would not be best solution. In general, keep the content of each page of your site unique.
Canonical Tag is used like this.
<link rel="canonical" href="http://www.example.com/" />
This resource on content duplication might give you an insight if you are a newbie..