I know this is old, but thought I'd point out that you also need to copy the public key to the second client
(or recompute it with ssh-keygen -y -f ~/.ssh/id_rsa_.. > ~/.ssh/id_rsa...pub)
- Public Key Authentication Method: "publickey"
The only REQUIRED authentication 'method name' is "publickey"
authentication. All implementations MUST support this method;
however, not all users need to have public keys, and most local
policies are not likely to require public key authentication for all
users in the near future.
With this method, the possession of a private key serves as
authentication. This method works by sending a signature created
with a private key of the user. The server MUST check that the key
is a valid authenticator for the user, and MUST check that the
signature is valid. If both hold, the authentication request MUST be
accepted; otherwise, it MUST be rejected. Note that the server MAY
require additional authentications after successful authentication.
Your ssh client begins the authentication by sending the public key (the signature referenced in bold above) to the server. The server, if the public key is an authorized key, sends a random session ID back to your client. Your client then encodes that session ID with the private key and sends that back to the server. The server decodes that session ID using the public key, and if it matches the original session ID, then authenticates your client.