That's because when you save the file to the other server, you're creating it on that server and it's that server's operating system that sets the last-modified time of the file (to whatever the time was when the file appeared there). The time it was last modified on your system is not included in the stream of bytes that gets transferred between the two computers.
Messing with the modification time could cause grief for any backup procedures the server's administrator employs; it may be better to store such informatio (if it really is necessary) in some other way.
You might bundle the files into a zip archive, transfer that, and unzip it on the other machine; archive files typically record modification time, and can reinstate it on the files it unpacks. Obviously, you'll need to be able to run such a program on the other server.
There may be a way to readjust file modification time remotely via ftp, but I certainly can't think of it right now.