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If we store the conversations at a random key, we want to hide who is accessing which key.

a server will be able to see who accesses which keys, likely revealing who is talking to whom else

It's possible to even hide that information, for that you'd need to take a look at Private Information Retrieval (PIR),

make use of the Diffie-Hellman protocol: with Alice and Bob having an asymmetric keypair, they can establish a secret that only the two know. This secret can be the "key" where a server could store a bunch of ciphertexts for them. This secret can be the "key" where a server could store a bunch of ciphertexts for them.

That said, a server will be able to see who accesses which keys, likely revealing who is talking to whom else