If you're only the editor (as opposed to the administrator) your solution may be in creating (by the administrator) a dummy sister-review-journal for these conflict of interest reviews, to which you would not have the access. See how I use a 'dummy journal' to provide an open public double-blind referring system with anonymous editors, authors, and referees (see http://mcfns.com/index.php/Blind
) for the actual journal at mcfns.com where everything becomes transparent.
In this kind of scenario you would submit your manuscript to your main journal but an editor in charge of the manuscript would copy it for the review processing into the sister-journal, to which you would not have the access while he or she would - it takes just a few minutes to move a submission between two journals. You could even have a few 'sister-journals' if you had a few editors. I have a separate installation of OJS system for administrating reviews for a journal that does not use the OJS software at all, and I am just temporarily the EiC for it in addition to having my own OJS based journal.
Of course, if you are the system administrator, or even worse if you are the server administrator who have the access to the MySQL installation, then there is no other way than finding another system administrator working on another server who would make an independent OJS installation - there is nothing the OJS programmers could do to prevent a system administrator from finding all the data in the database. A good reason for having 'mirror sites' is testing new ideas relating to your journal changes and backups - I have a few of those and consider them very useful (e.g., main.mcfns.com and quantfor.com).