What kind of index of papers is behind the search functionality of web applications and services related to the scientific literature?
There are various APIs that other services can use to search assorted indices from other sites 

For example, according to the about page of ShortScience.org: 


  The search can access all papers in the DOI, arχiv, dblp, and Bibsonomy databases


Some of those (e.g. DBLP) only host bibliographic data; others (e.g. arXiv) host the full text of articles or preprints. Some allow access to their data by API, others make their complete database available for download.

SciRate uses arXiv. Brevy does not seem to have an index; it's just a wiki, and the search function just searches user-submitted content.