TL;DR: TLS can in theory be used without private keys.
An integral part of TLS is that the client properly authenticates the server in order to detect potential man in the middle attacks. This is commonly done with certificates - and this requires the private key of the certificate on the server site to prove to the client that the server is the owner of the certificate1.
But, TLS is flexible and also provides other ways to authenticate the server. There are PSK ciphers which allow authentication by using a pre-shared key between client and server. There are SRP ciphers which allow authentication by password. There are even anon ciphers which allow you to abandon any kind of in-TLS authentication, i.e. require you to do your own authentication.
Thus, TLS might be used without certificates. And since certificates are the only part of TLS which requires a private key this means that TLS can be used without private keys. But, even if no private key is used a similar secret thing might be required on the server: both SRP and PSK based authentication require a secret on the server side which should at most be known to server and client.
Apart from this, certificate-less (and thus private-key-less) authentication requires both sides of the connection to support the relevant ciphers with the alternative authentication methods. Since browsers currently only support certificate-based authentication a private key is needed on the server side whenever access by a browser is required - i.e. practically every time in case of web servers.
For example you can use openssl to provide a TLS protected connection which does not require certificates. The server would look for example like this:
$ openssl s_server -www -nocert -psk 1a2b3c4d
The client would need to use the same key when connecting to the server, i.e. the basic idea behind PSK is that client and server share a secret key:
$ openssl s_client -connect 127.0.0.1:4433 -psk 1a2b3c4d
Cipher : PSK-AES256-CBC-SHA
Verify return code: 0 (ok)
If you would try to connect with a browser to this server (i.e.
https://127.0.0.1:4433) you would only get an error since the browser does not support the PSK ciphers:
1 Private keys can not only be required for authentication but also for key exchange, i.e. RSA key exchange. But, RSA key exchange requires certificates so it again means that private keys are only relevant if certificates are used. Also, the Diffie-Hellman key exchange does not need certificates and this is actually the recommended method today since it provides forward secrecy.