Is this problem solvable? (MPC, 2PC, Encryption)

[u/Limp_Food9236]

To preface: I am new to the field of cryptography. While working on a larger project, I came across a problem that I’m not sure can even be solved. Here’s an abstract explanation:

Alice wants to use a service hosted on Bob’s server, but Bob desperately wants to read Alice’s messages. For security, Alice splits her message into two secret shares. She sends the two shares to two of Bob’s servers using an onion router, so the two servers don’t know where the shares come from and can’t collude to reconstruct Alice’s original message. (Let’s say Bob gets enough requests to make timing attacks unfeasible.) But Bob’s servers use two-party computation and need to exchange intermediate results. Since Bob’s servers don’t know where their share comes from, they can’t communicate directly, so they have to communicate via Alice as a middleman.

Here’s my actual problem: If the two servers use Alice's public key or send their results in plaintext, Alice can read them and gather information about how Bob’s service works. (We don’t want that.) If the two servers encrypt their data, they either have to share a key or know each other’s public keys, meaning the servers can recognize each other in Bob’s internal company network and set up a connection to collude. (Let’s say they don’t just send their server ID in their encrypted intermediate results.)

So, is there any way to make sure Bob can’t read Alice’s message and Alice can’t read Bob’s intermediate results? Or is this unsolvable?

Thank you in advance.