No, file permissions.
then kill it and serve them?
Users with uid != 0 cannot kill other user's processes.
Or could it even just MITM it?
No, host key verification ensures that you are talking to the intended ssh daemon without packets being intercepted. The host key is a public key of the SSH server, which is verified by the client to be the key registered for that particular host. Since only the server has corresponding private key, the MITM cannot eavesdrop on the key exchange.
(I might be wrong about this, but...) I've seen quite a few instances where SSH refuses to let people even try to log in if there's a chance that a private key has the wrong permissions applied, so as to minimize the risk of someone accidentally letting others read it -- as you suggested.
You'll find the default SSH config on the more noteworthy Linux distributions to do this 90% of the time time without even asking :)
* Note, I guess you could run an sshd that only allows logging in to a single user and runs as that user but both points above still stand even in that case.