SQLite 3.27

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WITH yourTable AS (
    SELECT 1 AS row_id, 'Incoming' AS Type, '2020-08-10 08:09:18' AS Date, 'Hi' AS Text UNION ALL
    SELECT 2, 'Outgoing', '2020-08-10 08:11:04', 'Hello' UNION ALL
    SELECT 3, 'Incoming', '2020-08-10 08:11:12', 'For tomorrow' UNION ALL
    SELECT 4, 'Incoming', '2020-08-10 08:11:20', 'Are we still on for dinner?' UNION ALL
    SELECT 5, 'Outgoing', '2020-08-10 08:11:31', 'Let me check.' UNION ALL
    SELECT 6, 'Outgoing', '2020-08-10 08:11:43', 'Yes' UNION ALL
    SELECT 7, 'Incoming', '2020-08-10 08:11:45', 'Great!'
),
cte AS (
    SELECT *, ROW_NUMBER() OVER (ORDER BY Date) rn1,
              ROW_NUMBER() OVER (PARTITION BY Type ORDER BY Date) rn2
    FROM yourTable
),
cte2 AS (
    SELECT *, rn1-rn2 AS grp, DENSE_RANK() OVER (PARTITION BY Type ORDER BY rn1-rn2) rnk
    FROM cte
    ORDER BY grp, rnk
)

SELECT
    GROUP_CONCAT(CASE WHEN Type = 'Incoming' THEN TEXT END, ' ') AS Incoming,
    GROUP_CONCAT(CASE WHEN Type = 'Outgoing' THEN TEXT END, ' ') AS Outgoing
FROM cte2
GROUP BY
    rnk
ORDER BY
    rnk;

Incoming	Outgoing
Hi	Hello
For tomorrow Are we still on for dinner?	Let me check. Yes
Great!	null