This post is old, but I still will give an answer since recently I was having this issue and arrived to this post, this may be useful to someone.
As Andrew Schulman has answered, you would need to login with user/password to be granted the full Windows account privileges (For example: Admin rights if the user has those).
However, if you still want to use public key auth for security reasons (better security than standalone password auth) the following can be a solution:
Since ssh supports using more than one authentication method chained one after another (https://security.stackexchange.com/questions/17931/possible-to-use-both-private-key-and-password-authentication-for-ssh-login) you could enable both public key auth and password auth:
The purpose of public key auth would be to have stronger security.
The purpose of password auth would be to 'force' sshd service to create a Windows session with your full privileges by supplying it with the user/pass.
To enable this you would need to configure the following setting on your sshd_config file on the server with cygwin installed:
Please note that you also need the following enabled:
After the configuration file change and sshd service restart you will need both the private key and to enter the user password to login with ssh from client. You will be given the full Windows privileges of the specified account.
The small inconvenient is that in addition to the private key you will need to specify the user password every time you want to login with ssh, but I think this is still good for the sake of having better security and being able to have the full privileges of your Windows account.