Return a column per date in a range
You can use the PIVOT function to perform this query.  My answer will include both a Static and dynamic version because sometimes it is easier to understand it using a static version.

A Static Pivot is when you hard-code all of the values that you want to transform into columns.

-- first into into a #temp table the list of dates that you want to turn to columns
;with cte (datelist, maxdate) as
(
    select min(arrivaldate) datelist, max(departuredate) maxdate
    from BookingsPerPerson
    union all
    select dateadd(dd, 1, datelist), maxdate
    from cte
    where datelist &lt; maxdate
) 
select c.datelist
into #tempDates
from cte c

select *
from
(
    select b.person_id, b.arrivaldate, b.departuredate,
        d.datelist,
        convert(CHAR(10), datelist, 120) PivotDate
    from #tempDates d
    left join BookingsPerPerson b
        on d.datelist between b.arrivaldate and b.departuredate
) x
pivot
(
    count(datelist)
    for PivotDate in ([2012-01-01], [2012-01-02], [2012-01-03],
              [2012-01-04], [2012-01-05], [2012-01-06] , [2012-01-07])
) p;


Results (See SQL Fiddle With Demo):

PERSON_ID | ARRIVALDATE | DEPARTUREDATE | 2012-01-01 | 2012-01-02 | 2012-01-03 | 2012-01-04 | 2012-01-05 | 2012-01-06 | 2012-01-07
=====================================================================================================================================
123456    | 2012-01-01  | 2012-01-04    | 1          | 1          | 1          | 1          | 0          | 0          | 0
213415    | 2012-01-02  | 2012-01-07    | 0          | 1          | 1          | 1          | 1          | 1          | 1


The dynamic version will generate the list of values to transform to columns:

DECLARE @cols AS NVARCHAR(MAX),
    @query  AS NVARCHAR(MAX)

;with cte (datelist, maxdate) as
(
    select min(arrivaldate) datelist, max(departuredate) maxdate
    from BookingsPerPerson
    union all
    select dateadd(dd, 1, datelist), maxdate
    from cte
    where datelist &lt; maxdate
) 
select c.datelist
into #tempDates
from cte c


select @cols = STUFF((SELECT distinct ',' + QUOTENAME(convert(CHAR(10), datelist, 120)) 
                    from #tempDates
            FOR XML PATH(''), TYPE
            ).value('.', 'NVARCHAR(MAX)') 
        ,1,1,'')

set @query = 'SELECT person_id, arrivaldate, departuredate, ' + @cols + ' from 
             (
                select b.person_id, b.arrivaldate, b.departuredate,
                    d.datelist,
                    convert(CHAR(10), datelist, 120) PivotDate
                from #tempDates d
                left join BookingsPerPerson b
                    on d.datelist between b.arrivaldate and b.departuredate
            ) x
            pivot 
            (
                count(datelist)
                for PivotDate in (' + @cols + ')
            ) p '

execute(@query)


The results are the same (see SQL Fiddle With Demo):

PERSON_ID | ARRIVALDATE | DEPARTUREDATE | 2012-01-01 | 2012-01-02 | 2012-01-03 | 2012-01-04 | 2012-01-05 | 2012-01-06 | 2012-01-07
=====================================================================================================================================
123456    | 2012-01-01  | 2012-01-04    | 1          | 1          | 1          | 1          | 0          | 0          | 0
213415    | 2012-01-02  | 2012-01-07    | 0          | 1          | 1          | 1          | 1          | 1          | 1