INSERT New Record Works OK In Local Table, BUT Not If The Target SS DB/table Is In A Different Physical Server

Hi... I was hoping if someone could share me some thoughts with the issue that I am having at the moment.

Problem: When I run the package in my local machine and update local SS DB/table - new records writes OK in the table. BUT when I changed my destination meaning write record into another physical SS DB/table there is no INSERT data occurs. AND SO when I move/copy over that same package into another server (e.g. server that do not write record earlier) and run it locally IT WORKS fine too.

What I am trying to do is very simple - Add new records in a SS table using SSIS . I only care for new rows and not even changed rows.
Here is my logic -
1. Create Ole DB source to RemoteSERVER - using SELECT stmt
2. I have LoopUp component that will look for NEW records - Directs all rows that don't find match and redirect rows (error output).
3. Since I don't care for any rows that is matched in my lookup - I do nothing or I trash the rows
4. I send the error rows (NEW rows) into OleDB destination

RESULTS when I run the package locally and destination table is also local - WORKS FINE;
But when I run the package locally and destination table is in another Sserver (remote) - now rows is written.

The package is run thru BIDS manually so there is no sucurity restrictions attached to it.

I am not sure what I am missing. And I do not see error in my package either. It is not failing.

Thanks in advance!