If you are able to wrap your XML in a root element - say then the following is your solution:
DECLARE @PersonsXml XML = '<persons><person><firstName>Jon</firstName><lastName>Johnson</lastName></person>
<person><firstName>Kathy</firstName><lastName>Carter</lastName></person>
<person><firstName>Bob</firstName><lastName>Burns</lastName></person></persons>'
SELECT b.value('(./firstName/text())[1]','nvarchar(max)') as FirstName, b.value('(./lastName/text())[1]','nvarchar(max)') as LastName
FROM @PersonsXml.nodes('/persons/person') AS a(b)