Is it possible for a Mumble server operator to monitor or record users' conversations? Or are Mumble conversations encrypted end-to-end?
With public-key cryptography, the private key is always known by the server. Thus, a Mumble server administrator indeed has access to the private key - usually in the form of a X.509 certificate - which can be used to decrypt the TLS handshake to derive a session's symmetric cipher key. This key is used to encrypt and decrypt all data transmitted over the connection. In other words, an administrator certainly is capable of decrypting all traffic.
Also, nothing stops a server administrator to log and record communication on the server.
Adding to Stevens correct answer: http://mumble.sourceforge.net/FAQ#Is_Mumble_encrypted.3F chat is indeed TLS encrypted and voip via OCB-AES but both channels are only server-client as opposed to end-to-end, i.e. the server (operator) has in principle access to all clear text and voice running through the server.
Hence I run my own server (setup is really easy), see http://mumble.sourceforge.net/Running_Murmur