SSH key authentication with another user

You need to populate the ~/.ssh/authorized_keys file on Host B with the public keys of all the users you want to have access to the shared account. Only the client needs the user's private key. The server only needs to know the public keys which are allowed to connect to a given account.

There is rarely a need for users to share private keys (the well-known Vagrant private key is an illustrative exception), or to place the users' private keys on the server. Each private key should be unique to each user and should remain on his or her client machine. Even if you disregard this best practice, you still don't need to put the shared-user's private key on the server; you just need to distribute a copy of the private key to every user who will be using that key to connect to the shared account.