A user needs to have access to a table in order to perform any action on it.
Make sure that you have the appropriate roles defined, and that privileges are revoked from the PUBLIC role.
Grant access to the schema containing the table, as follows:
GRANT USAGE ON SCHEMA someschema TO somerole;
GRANT SELECT, INSERT, UPDATE, DELETE ON someschema.sometable TO somerole;
GRANT somerole TO someuser, otheruser;