When you do this:
git clone git@gitserver:proj/myrepo.git
an ssh client is starting on the local host ('sshserver') and authenticating with 'gitserver' using public key authentication. If you get prompted for a password for user 'git', that means public key authentication has failed, and ssh is falling through to the next method, which is password authentication.
The most likely reason that public key authentication would fail is that the ssh client does not have the private key needed. I suspect in this case that the key needed to authenticate as 'git@gitserver' resides in sshserver:~/.ssh, in which case it would not be available to the ssh client started on your local host when you try to clone the repo through your ssh tunnel.
To solve this, you need to give that client access to the appropriate key. You could add it to ~/.ssh locally, or load it into an ssh agent.