How to implement business logic permissions in PostgreSQL (or SQL in general)?
I know that you are not asking about database security per se, but you can do what you want using database security. You can even use this in a web app. If you don't want to use database security, then the schemas still apply. 

You want column-level security, row-level security, and probably hierarchical role management. Role-Based security is much easier to manage than User-Based security. 

This example code is for PostgreSQL 9.4, which comes out soon. You can do it with 9.3, but there is more manual labour required.

You want everything to be indexable if you are concerned with performance &dagger;, which you should be. This means that bit-mask and array fields probably won't be a good idea.

In this example, we keep the main data tables in the data schema, and corresponding views in public. 

create schema data; --main data tables
create schema security; --acls, security triggers, default privileges

create table data.thing (
  thing_id int primary key,
  subject text not null, --or whatever
  owner name not null
);


Put a trigger on data.thing for inserts and updates enforcing that the owner column is the current_user. Perhaps allow only the owner to delete his own records (another trigger). 

Create a WITH CHECK OPTION view, which is what users will actually use. Try really hard to make it updatable, otherwise you'll need triggers/rules, which is more work.  

create view public.thing with(security_barrier) as 
select
thing_id,
subject,
owner,
from data.thing
where
pg_has_role(owner, 'member') --only owner or roles "above" him can view his rows. 
WITH CHECK OPTION;


Next, create an access-control list table:

--privileges r=read, w=write

create table security.thing_acl (
  thing_id int,
  grantee name, --the role to whom your are granting the privilege
  privilege char(1) check (privilege in ('r','w') ),

  primary key (thing_id, grantee, privilege),

  foreign key (thing_id) references data.thing(thing_id) on delete cascade
);


Change your view to account for ACLs:

drop view public.thing;

create view public.thing with(security_barrier) as 
select
thing_id,
subject,
owner
from data.thing a
where
pg_has_role(owner, 'member')
or exists (select 1 from security.thing_acl b where b.thing_id = a.thing_id and pg_has_role(grantee, 'member') and privilege='r')
with check option;


Create a default row privileges table:

create table security.default_row_privileges (
  table_name name,
  role_name name,
  privilege char(1),

  primary key (table_name, role_name, privilege)
);


Put a trigger on insert on data.thing so that it copies default row privileges to security.thing_acl . 


Adjust table-level security appropriately (prevent inserts from unwanted users). No one should be able to read the data or security schemas. 
Adjust column-level security appropriately (prevent some users from seeing/editing some columns). You can use has_column_privilege() to check that a user can see a column. 
Probably want security definer tag on your view. 
Consider adding grantor and admin_option columns to acl tables to track who granted the privilege, and whether the grantee can manage privileges on that row. 
Test lots


&dagger; In this case pg_has_role is probably not indexable. You would have to get a list of all superior roles to current_user and compare to the owner/grantee value instead.