Should we then provide the defense access to the prosecution's evidence (and vice-versa)?
The issue goes beyond that, to providing the defense comparable ability to collect evidence to the prosecution. Giving the defense access to prosecution evidence doesn't help if the prosecution deliberately does not collect evidence that it believes would be exculpatory and the defense doesn't have comparable means to compel access to potential evidence.
The DA must follow this rule: http://codes.findlaw.com/ca/penal-code/pen-sect-1054-1.html.
The DA also has obligations under Brady v. Maryland: https://scholar.google.com/scholar_case?case=955043312626967...
The defense must follow this rule: http://codes.findlaw.com/ca/penal-code/pen-sect-1054-3.html