I recently created a Flow (called from a page layout button) and my functional test of it worked when logged in as Admin. But I ran into the "Insufficient Privileges" error when testing it as a sales user. You know, this one:
You do not have the level of access necessary to perform the operation you requested. Please contact the owner of the record or your administrator if access is necessary.
All object and field permissions (the Flow performs DML) are the same between the two profiles for the relevant objects and fields.