BigQuery comparing DATE and TIMESTAMP

Below is for BigQuery Standard SQL

#standardSQL
SELECT
  IFNULL(OnSite.worksite_id, Documents.worksite_id) AS `Worksite`,
  IFNULL(OnSite.timestamp, DATE(Documents.timestamp)) AS `DATE`,
  COUNT(Documents.worksite_id) AS `Users_on_Site`,
  COUNT(DISTINCT OnSite.uid) AS `Completed`
FROM `project.dataset.OnSite` OnSite
LEFT JOIN `project.dataset.Documents` Documents 
ON OnSite.worksite_id = Documents.worksite_id 
AND OnSite.timestamp = DATE(Documents.timestamp)
GROUP BY `DATE`, `Worksite`

if to apply to sample data from your question

WITH `project.dataset.OnSite` AS (
  SELECT "u12345" uid, "worksite_1" worksite_id, DATE '2019-01-01' `TIMESTAMP` UNION ALL
  SELECT "u12345", "worksite_1", '2019-01-02' UNION ALL
  SELECT "u12345", "worksite_1", '2019-01-03' UNION ALL
  SELECT "u12345", "worksite_1", '2019-01-04' UNION ALL
  SELECT "u12345", "worksite_1", '2019-01-05' UNION ALL
  SELECT "u12345", "worksite_1", '2019-01-06' UNION ALL
  SELECT "u1", "worksite_1", '2019-01-01' UNION ALL
  SELECT "u1", "worksite_1", '2019-01-02' UNION ALL
  SELECT "u1", "worksite_1", '2019-01-05' UNION ALL
  SELECT "u1", "worksite_1", '2019-01-06' 
), `project.dataset.Documents` AS (
  SELECT "1" document_id,     "u12345" uid,   "worksite_1" worksite_id, 'work_permit' type,    TIMESTAMP '2019-01-01 00:00:00' `TIMESTAMP` UNION ALL
  SELECT "2",     "u12345",   "worksite_2", 'job',            '2019-01-02 00:00:00' UNION ALL
  SELECT "3",     "u12345",   "worksite_1", 'work_permit',    '2019-01-03 00:00:00' UNION ALL
  SELECT "4",     "u12345",   "worksite_2", 'job',            '2019-01-04 00:00:00' UNION ALL
  SELECT "5",     "u12345",   "worksite_1", 'work_permit',    '2019-01-05 00:00:00' UNION ALL
  SELECT "6",     "u12345",   "worksite_2", 'job',            '2019-01-06 00:00:00' UNION ALL
  SELECT "7",     "u12345",   "worksite_1", 'work_permit',    '2019-01-07 00:00:00' UNION ALL
  SELECT "8",     "u12345",   "worksite_2", 'work_permit',    '2019-01-09 00:00:00' UNION ALL
  SELECT "9",     "u12345",   "worksite_1", 'job',            '2019-01-09 00:00:00' UNION ALL
  SELECT "10",    "u12345",   "worksite_2", 'work_permit',    '2019-01-09 00:00:00' UNION ALL
  SELECT "11",    "u12345",   "worksite_1", 'work_permit',    '2019-01-09 00:00:00' UNION ALL
  SELECT "12",    "u12345",   "worksite_2", 'work_permit',    '2019-01-09 00:00:00' UNION ALL
  SELECT "13",    "u12345",   "worksite_1", 'job',            '2019-01-09 00:00:00' UNION ALL
  SELECT "14",    "u12345",   "worksite_2", 'work_permit',    '2019-01-09 00:00:00' UNION ALL
  SELECT "15",    "u12345",   "worksite_1", 'work_permit',    '2019-01-09 00:00:00' 
)

result will be as expected

Row Worksite    Date        Users_on_Site   Completed    
1   worksite_1  2019-01-01  2               2    
2   worksite_1  2019-01-02  0               2    
3   worksite_1  2019-01-03  1               1    
4   worksite_1  2019-01-04  0               1    
5   worksite_1  2019-01-05  2               2    
6   worksite_1  2019-01-06  0               2

In BigQuery documentation, it is explained that DATE function accepts following input :

DATE(year, month, day) : Constructs a DATE from INT64 values representing the year, month, and day.

DATE(timestamp_expression[, timezone]) : Converts a timestamp_expression to a DATE data type. It supports an optional parameter to specify a timezone. If no timezone is specified, the default timezone, UTC, is used.

In your use case, it seems like the value you are passing to DATE is already a datetime. For this purpose, you could use DATETIME_TRUNC, like :

DATETIME_TRUNC(IFNULL(OnSite.timestamp, Documents.timestamp), DAY)

BigQuery comparing DATE and TIMESTAMP

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