1/ How are privileges on synonyms and underlying objects related ? If one has rights on synonym, would he automatically has rights on the table and vice versa ?
Oracle says
When you grant object privileges on a synonym, you are really granting privileges on the underlying object, and the synonym is acting only as an alias for the object in the GRANT statement
which means privilege on synonym is enough. That will bypass table privilege.
Another source says that access right on table is enough and synonym privilege has no meaning.
Does it mean either privilege on the synonym or the underlying table is enough ?
2/ Is the behavior the same for private and public synonym. I haven't really seen an example of granting privileges on synonyms for a user to "see/access". How to grant privilege on private synonyms to a user ?
Both the Oracle docs and the message you referred to say exactly the same thing. Privileges are not granted on a synonym. When you attempt to grant privileges on a synonym the database actually performs the grant on the object referred to by the synonym. Thus, it makes no difference if the synonym is public or private because the actual grant is made on the object referred to by the synonym.
Best of luck.
Let's demonstrate what happens:
-- Logged in as user BOB2
CREATE TABLE RPJ_TEST (N NUMBER);
SELECT *
FROM DBA_TAB_PRIVS
WHERE TABLE_NAME = 'RPJ_TEST';
-- the above statement returns no rows
CREATE SYNONYM RPJ_TEST_SYN -- create synonym
FOR RPJ_TEST;
SELECT *
FROM DBA_TAB_PRIVS
WHERE TABLE_NAME = 'RPJ_TEST';
-- the above statement returns no rows
GRANT SELECT ON RPJ_TEST TO BOB; -- grant on table
SELECT *
FROM DBA_TAB_PRIVS
WHERE TABLE_NAME = 'RPJ_TEST';
-- the above statement returns
GRANTEE OWNER TABLE_NAME GRANTOR PRIVILEGE GRANTABLE HIERARCHY
BOB BOB2 RPJ_TEST BOB2 SELECT NO NO
GRANT UPDATE ON RPJ_TEST_SYN TO BOB2; -- grant "on synonym" actually performs grant on table
SELECT *
FROM DBA_TAB_PRIVS
WHERE TABLE_NAME = 'RPJ_TEST';
GRANTEE OWNER TABLE_NAME GRANTOR PRIVILEGE GRANTABLE HIERARCHY
BOB BOB2 RPJ_TEST BOB2 SELECT NO NO
BOB BOB2 RPJ_TEST BOB2 UPDATE NO NO
Note that after the grant on the synonym RPJ_TEST_SYN the privileges granted on the table referred to by the synonym had been changed.