Replace your Elastic IP address with the private IP address. Then configure you, security groups, in a way that the instances can communicate.
Consider the two instances as Instance-1 and Instance-2
I am running this code on Instance1
scp -i ~/Path-To-Key-File/yourkeypair.pem /path/file [email protected]<Private IP of Instance-2>:/path/file
In Instance-2, allow SSH protocol over private IP to communicate.