The directory /etc/ssh/ contains shared, global files for the all the users on that machine regardless of whether for the SSH client or SSH server. The files in ~/.ssh/ are specific to a particular user, again both client and server files. But that is just the default location, you can move things around especially on the server with directives like AuthorizedKeysFile in sshd_config.
See the tail end of the manual pages for both the SSH client and SSH server for information about the files.
The keys you find there by default in /etc/ssh/ are only those the server uses to identify itself when you connect to it. If you have the right server, as determined by matching server keys, then authentication goes ahead with user keys or passwords.
You can put user keys there, too, or in a subdirectory so that users can still authenticate with keys despite having encrypted home directories. There you have more choice about whether the users can or can't update their own keys.