If you install Web Deploy using the Web Platform Installer (WPI) within IIS, some components wont be added by default and you wont be able to make it work.
You will need to open incoming TCP port 8172 in your firewall (if any).
Using the WPI, install all three:
Then open: Control Panel > Programs > Programs and Features
Find Microsoft Web Deploy 3.5 and select "Change".
In the following wizard chose "Change" again and install the IIS Deployment Handler.
The Deployment Handler wont be listed unless "Web Deploy 3.5 for Hosting Servers" is installed.
If the Deployment Handler is not installed the context menu in IIS wont appear.
You should now be able to grant publishing permission to a user.