There does not seem to be a built-in way of editing files which are owned by other users in Emacs.
The two ways that most people seem to be doing it are: via "tramp" or using "sudoedit".
Here is one example:
alias E="SUDO_EDITOR=\"emacsclient -c -a emacs\" sudoedit"
With tramp (see discussion here):
It is possible to edit files like this:
C-c C-f /sudo::/path/to/file
There is also an Emacs package called "sudo-save" available.
I find it surprising that something like sudo-save hasn't been made available in distributions of Emacs. It might of course have to do with security reasons. I have yet to find an elegant solution. Ideally I would prefer if, when opening files owned to another user, Emacs would ask me if I would want to edit using sudo or in read-only mode.