The optimization process and recommendations:

1. Avoid Using Date Functions In Conditions (query line: 16): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. An alternative way is to use a range condition instead of a function call.
2. Avoid Using Date Functions In Conditions (query line: 22): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. An alternative way is to use a range condition instead of a function call.
3. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
4. Replace In Subquery With Correlated Exists (modified query below): In many cases, an EXISTS subquery with a correlated condition will perform better than a non correlated IN subquery.

Optimal indexes for this query:

CREATE INDEX loan_idx_id_sender_id_receiver ON Loan (id_sender,id_receiver);

The optimized query:

SELECT
        User.name 
    FROM
        User 
    WHERE
        EXISTS (
            SELECT
                1 
            FROM
                Loan 
            WHERE
                (
                    Loan.id_receiver IN (
                        SELECT
                            DISTINCT id_receiver 
                        WHERE
                            date BETWEEN '2014-01-01 00:00:00' AND '2014-12-31 23:59:59'
                    ) 
                    AND Loan.id_receiver IN (
                        SELECT
                            DISTINCT id_receiver 
                        WHERE
                            date BETWEEN '2015-01-01 00:00:00' AND '2015-12-31 23:59:59'
                    )
                ) 
                AND (
                    User.id = Loan.id_sender
                )
            )