How can I get a trigger to fire on each inserted row during an INSERT INTO Table (etc) SELECT * FROM Table2?

The insert trigger is called once for bulk inserts, but on the trigger you can use the special inserted table to get all the inserted rows.

So, imagine you have an insert trigger like this one, that logs all the rows inserted into table

create trigger trgInsertTable 
on dbo.table
for insert
as
   insert tableLog(name)
    select name from inserted

With this trigger, when you make a bulk insert on table, the tableLog is filled with the same number of rows that were inserted to table

For you specific trigger, since you need to call a stored procedure for each row, you need to use a cursor:

ALTER TRIGGER dbo.Notify
    ON  dbo.Table
    AFTER INSERT
AS 
BEGIN

    DECLARE @EmailSender varchar(50)='Sender Profile'
    DECLARE @User varchar(20)
    DECLARE @Subject varchar(50)

    DECLARE cursor CURSOR FOR
      SELECT User, '(' + CONVERT(varchar, Id) + ')!'
        FROM inserted

    OPEN cursor
    FETCH NEXT FROM cursor INTO @User, @Subject
    WHILE @@FETCH_STATUS = 0
    BEGIN
      exec msdb.dbo.sp_send_dbmail
          @profile_name=@EmailSender,
          @recipients=@User
          @subject=@Subject,
          @body='//etc'
      FETCH NEXT FROM cursor INTO @User, @Subject
    END
    CLOSE cursor
    DEALLOCATE cursor

END

I didn’t tested, but should work