I had the same problem on MS SQL Server 2000 Service Pack 3a, and honestly believe that it is a glitch/bug. I suspect it has something to do with permissions ... that I have different account names on one machine from the next.
After changing every single option, and having it still not populate the tables, I created my own DTS package, and did everything manually.
For my purposes, I did not need to have the id field auto-increment on the destination server, nor did I need to copy any triggers, indexes, etc. ... I just needed the data there.
So, I created 3 objects:
1. SQL Command: DELETE FROM tablename
This cleared out the table of all data
2. SERVER object to SOURCE server
3. SERVER object to DESTINATION server
I then setup a data transformation between the two servers, and straight up copied the table from one to the other. I added a workflow line from the SQL command, so that on SUCCESS, it would then run the data transformation between the servers.
Again, this is rather simplistic, and won't copy triggers, indexes, etc., just the data.
Good luck!