Your example of
rsync -av /home/ME/myfile user@remoteserver
has left off the colon. Without that, it assumes the file is local.
If you instead had
rsync -av /home/ME/myfile user@remoteserver:
Then it would make a remote copy. In this case the location is relative to the home directory of the user. With no relative path given, the file will appear in that directory.
$ rsync -v myfile user@remoteserver:
sent 619 bytes received 35 bytes 145.33 bytes/sec
total size is 529 speedup is 0.81
$ ssh user@remoteserver "ls ./myfile"