Есть два идентичных сервера. На один пускает по ключу. На второй просит пароль. Ключи одинаковые везде.
ssh <User>@x.x.x.x -v
OpenSSH_6.7p1 Debian-5, OpenSSL 1.0.1k 8 Jan 2015
debug1: Reading configuration data /etc/ssh/ssh_config
debug1: /etc/ssh/ssh_config line 19: Applying options for *
debug1: Connecting to x.x.x.x [x.x.x.x] port 22.
debug1: Connection established.
debug1: identity file /home/<User>/.ssh/id_rsa type 1
debug1: key_load_public: No such file or directory
debug1: identity file /home/<User>/.ssh/id_rsa-cert type -1
debug1: Enabling compatibility mode for protocol 2.0
debug1: Local version string SSH-2.0-OpenSSH_6.7p1 Debian-5
debug1: Remote protocol version 2.0, remote software version OpenSSH_5.3
debug1: match: OpenSSH_5.3 pat OpenSSH_5* compat 0x0c000000
debug1: SSH2_MSG_KEXINIT sent
debug1: SSH2_MSG_KEXINIT received
debug1: kex: server->client aes128-ctr umac-64@openssh.com none
debug1: kex: client->server aes128-ctr umac-64@openssh.com none
debug1: SSH2_MSG_KEX_DH_GEX_REQUEST(1024<3072<8192) sent
debug1: expecting SSH2_MSG_KEX_DH_GEX_GROUP
debug1: SSH2_MSG_KEX_DH_GEX_INIT sent
debug1: expecting SSH2_MSG_KEX_DH_GEX_REPLY
debug1: Server host key: RSA 8e:41:f5:a1:f2:ae:39:a0:0a:1d:f3:99:5a:71:1f:01
debug1: Host 'x.x.x.x' is known and matches the RSA host key.
debug1: Found key in /home/<User>/.ssh/known_hosts:3
debug1: SSH2_MSG_NEWKEYS sent
debug1: expecting SSH2_MSG_NEWKEYS
debug1: SSH2_MSG_NEWKEYS received
debug1: Roaming not allowed by server
debug1: SSH2_MSG_SERVICE_REQUEST sent
debug1: SSH2_MSG_SERVICE_ACCEPT received
debug1: Authentications that can continue: publickey,gssapi-keyex,gssapi-with-mic,password
debug1: Next authentication method: publickey
debug1: Offering RSA public key: /home/<User>/.ssh/id_rsa
debug1: Authentications that can continue: publickey,gssapi-keyex,gssapi-with-mic,password
debug1: Next authentication method: password
Даже если положить публичный ключ для рута - рута пускает без пароля по этому ключу, а пользователя нет.