I think you're backwards. Or maybe I am. Referring to two servers rather than a server and a client makes this more confusing; in any ssh connection, one side is acting as a server and the other side is acting as a client, no matter what other purpose the two machines have. When using public-key authentication, the ssh server knows the public half of the key, and the ssh client knows the private half of key (and also the public half). If the key is vulnerable, then any client given a bunch of tries can guess the private half of the key.
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