The trouble is, FileMaker enforces its record-level security across tables and files (otherwise it wouldn't be security). You cannot get around it by doing something like creating a calculation in a 2nd table, which points back to a field in the 1st table (where the user cannot View that data). The user may have access to the local data, but a relationship to the 1st table is only going hit records where he has access, not to other records which would otherwise match; because the user does not have access to even the ID field of those other records. Hence you're no better off than you were with one table.
I think that the only way to get what you want would be to create a "shadow table", which duplicated, and was synchronized to, the data of the original table; but had only the fields you wanted generally exposed. Doable, but a PITA, especially the synchronization. It may be worth doing for only a few fields; but if a lot,...
If they are allowed to view the "other" records under some circumstances, then there seems no point in denying access in Accounts & Privileges. Looks like you could solve the entire thing purely through user interface.