The optimization process and recommendations:

1. 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.
2. 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:

ALTER TABLE `conversation_users` ADD INDEX `conversation_users_idx_conversation_id_user_id` (`conversation_id`,`user_id`);

The optimized query:

SELECT
        conversation_users.user_id,
        conversation_users.conversation_id 
    FROM
        conversation_users 
    WHERE
        EXISTS (
            SELECT
                1 
            FROM
                conversation_users AS conversation_users1 
            WHERE
                (
                    conversation_users1.user_id = 1
                ) 
                AND (
                    conversation_users.conversation_id = conversation_users1.conversation_id
                )
        )