My thoughts.
It is up to the institution/investigator as to exactly where the trail data, on consented subjects, is stored. Interestingly ICH E6 R2 states that:- "The investigator should maintain adequate and accurate source documents and trial records that include all pertinent observations on each of the site’s trial subjects" (4.9.0). Clinical trial activity begins with consent and as a consequence the subject allows the investigator to begin trial related screening processes and record trial related data. Data from consented subjects should be retained in the clinical trial documentation.
Obviously to comply with GCP, the data has to be readily available for monitoring, auditing and Inspection. In fact ICH E6 R2 says that the location of such documents should be documented:- "The sponsor and investigator/institution should maintain a record of the location(s) of their respective essential documents including source documents. (8.1).

ICH E3 also expects that screen failures are accounted for in the clinical study report.

RQA (Research Quality Association) answered a similar question about ID and screening logs, in October 2015. You might find the part about why data on screen failures is retained, useful:--
"........ Complying with ICH E6 (R1) 8.3.21, the investigator/institution needs to be able to reveal the identity of any subject. In the case of Screen Failures (SF) this may be for a variety of reasons, for instance:-
• Inspectors, auditors & monitors need to access the identity of all subjects, including screen failures, so they can locate pertinent information regarding their suitability to be enrolled in screening, ensuring their rights are safeguarded, etc;
• Data Protection (DP), Privacy & Confidentiality considerations:- so that subjects can be fully protected. The investigator & staff (and monitors) can't protect the information on subjects if they can't identify them. Subjects might want to file a DP related complaint and having an ID list will facilitate identifying the pertinent information;
• Well being and rights considerations:- SF subjects might have cause to complain about the tests and procedures administered during screening (even psychological tests) and hence it would be necessary to identify these subjects;
• Some sponsors routinely collect SAE data from any subject that gives consent. Hence verification and follow-up of SAE information would require identification of all subjects."

The MHRA GCP Guide (2012) has lots of useful information. 10.7.1, 11.4.1, 11.5.1 and others, might be useful