Ah yes, that is how it is intended in SQL. You get the Max of every column seperately. It seems like you want to return values from the row with the max date, so you have to select the row with the max date. I prefer to do this with a subselect, as the queries keep compact easy to read.
SELECT TrainingID, CompletedDate, Notes
FROM HR_EmployeeTrainings ET
WHERE (ET.AvantiRecID IS NULL OR ET.AvantiRecID = @avantiRecID)
AND CompletedDate in
(Select Max(CompletedDate) from HR_EmployeeTrainings B
where B.TrainingID = ET.TrainingID)
If you also want to match by AntiRecID you should include that in the subselect as well.