Steps to reproduce:
1. Enable local staging for a site, leave checked only Web Content. So documents are not staged, which means they will be stored and indexed only for the live group ID.
2. Create a document folder and upload a couple of files into it.
3. Create a web content structure with a Documents and Media field. Create also a basic template for it.
4. Create an article based on this structure. Click on select to browse for a document.
Result: when you browse around you can see the documents, but if you search for one, it does not find it
When you explicitly search in that popup document browser, it erroneously searches in the staging site, it does not take into consideration the fact that documents are not staged, so there is no two versions of each (a staging and a live), but only one version, and that is stored for the live site. So in this case it should rather search by the live groupid.