The best way to approach this would be to create a new key for the user you want on the Server1 to be connecting to Server2, using ssh-keygen, and copying that user's PUBLIC key into
~/.ssh/authorized_keys of Server2, then repeating the same procedure for the reverse trip.
YOUR private key must remain yours, it must never leave your PC, don't use it by copying it on the servers, I can assure you that you're not simplifying anything by doing that.
In all my CentOS 7 servers the default SSH configuration resides in ~/.ssh, the private key is usually
~/.ssh/id_rsa and the public is usually
these are the only 3 files you should need to create or edit for every machine:
ensure that the ~/.ssh directory is chmod'ed 700 and the files inside 600.
You can use
ssh-copy-id to send the public key from one server to another once you have created both sets of keys.