There's one setting for what the user can view and another that allows the user to edit. The edit option can be set to be based on a calculation that conditionally permits or prevents editing, such as one that compares the current account name to the account name used to create the record. (Select "limited..." in the edit column of custom privileges.)
See "Editing record access privileges" in FileMaker Help and check out this particular sub section: "Entering a formula for limiting access on a record-by-record basis" for a description of how to set this up.
I had been searching for exactly this and had no luck. Thanks for the tip! Wasn't able to reply earlier because I was on iOS.