You (as MATE user) do not need both files, but system needs both.
Below is some analysis:
-
polkit-gnome-authentication-agent-1.desktop
came from policykit-1-gnome
package (see 1).
-
polkit-mate-authentication-agent-1.desktop
came from mate-polkit-common
package (see 2)
On my Ubuntu MATE 16.04 LTS I have installed both.
Why? Let's ask Aptitude:
$ aptitude why policykit-1-gnome
i screen-resolution-extra Depends policykit-1-gnome
$ aptitude why screen-resolution-extra
i xserver-xorg Recommends xserver-xorg-video-all | xorg-driver-video
p nvidia-384 Provides xorg-driver-video
p nvidia-384 Recommends nvidia-settings (>= 331.20)
p nvidia-settings Depends screen-resolution-extra (>= 0.12)
$ aptitude why mate-polkit-common
i mate-polkit Depends mate-polkit-common (= 1.12.0-3)
$ aptitude why mate-polkit
i mate-desktop-environment-core Depends mate-polkit (>= 1.12.0)
So these files were installed as dependencies of Xorg server and MATE DE.
You should not remove them.
They operate in smart manner: only one is really active in one time - it is made by special directive of desktop-file. It is named OnlyShowIn
:
$ grep Only polkit-gnome-authentication-agent-1.desktop polkit-mate-authentication-agent-1.desktop
polkit-gnome-authentication-agent-1.desktop:OnlyShowIn=GNOME;XFCE;Unity;
polkit-mate-authentication-agent-1.desktop:OnlyShowIn=MATE;