SQL Server 2019

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DECLARE @Table TABLE(
RequestID varchar(100)
, PersonID varchar(100)
, StartDate DATE
, EndDate DATE
, StatusType VARCHAR(150)
);

INSERT INTO @Table
SELECT '445566', 'Person2', '9/13/2022', '9/16/2022', 'TransactionStarted'
UNION ALL
SELECT '445566', 'Person2', '9/13/2022', '9/16/2022', 'TransactionEnded'
UNION ALL
SELECT '112233', 'Person2', '8/13/2022', '8/16/2022', 'TransactionStarted'
UNION ALL
SELECT '112233', 'Person2', '8/13/2022', '8/16/2022', 'TransactionEnded'
UNION ALL
SELECT '556677', 'Person1', '8/10/2022', '8/12/2022', 'TransactionStarted'
UNION ALL
SELECT '556677', 'Person1', '8/10/2022', '8/12/2022', 'TransactionEnded';

SELECT COUNT(Distinct T.PersonID) TotalCountPerPerson, COUNT(Distinct CASE WHEN D.PersonID IS NULL THEN T.PersonID END) NewThisMonth,
       DATENAME(mm, T.StartDate) [SRMonthName], 
       MONTH(T.StartDate) [SRMonthNumber], YEAR(T.StartDate) [SRYear]
FROM @Table T LEFT JOIN @Table D
ON T.PersonID = D.personID AND T.StartDate > D.StartDate
GROUP BY MONTH(T.StartDate), DATENAME(mm,T.StartDate), YEAR(T.StartDate)
ORDER BY YEAR(T.StartDate), MONTH(T.StartDate);
/*--------------------------------------------------------------------*/

WITH CTE AS
  (
    SELECT *, CASE WHEN NOT EXISTS
                   (SELECT 1 FROM @Table D WHERE D.PersonID = T.PersonID AND D.StartDate<T.StartDate) 
                   THEN T.PersonID
              END AS ISNEW
    FROM @Table T 
  )
SELECT COUNT(Distinct PersonID) TotalCountPerPerson, 
       COUNT(Distinct ISNEW) NewThisMonth,
       DATENAME(mm, StartDate) [SRMonthName], 
       MONTH(StartDate) [SRMonthNumber], YEAR(StartDate) [SRYear]
FROM CTE
GROUP BY MONTH(StartDate), DATENAME(mm,StartDate), YEAR(StartDate)
ORDER BY YEAR(StartDate), MONTH(StartDate);

TotalCountPerPerson	NewThisMonth	SRMonthName	SRMonthNumber	SRYear
2	2	August	8	2022
1	0	September	9	2022

Warning: Null value is eliminated by an aggregate or other SET operation.

TotalCountPerPerson	NewThisMonth	SRMonthName	SRMonthNumber	SRYear
2	2	August	8	2022
1	0	September	9	2022

Warning: Null value is eliminated by an aggregate or other SET operation.