MySQL Table Schema Design - Relationship with Junction or User Table
Query 1:

SELECT them.user_id, them.first_name, them.last_name, them.email
    FROM user AS me
    JOIN user AS them  ON me.organization_id = them.organization_id
    WHERE me.user_id = ?


Query 2:

INSERT INTO notification (requester, receiver, ...)
    VALUES
    (?, ?, ...)


(I assume you have your id, his id, and other stuff to supply; do not supply notification_id, it will be generated.)

Query 3 is phrased imprecisely:

SELECT first_name, last_name, email,
       timestamp,   -- Don't you need a date&amp;time in the table?
       ...   -- don't you want more fields?
    FROM notifications
    WHERE email = ?  -- Leave out if you want all users; change to search by name
    ORDER BY `timestamp` DESC  -- I assume by 'current' you mean 'recent'?
    LIMIT 10;