I use Ubuntu 16.04 in English and Hebrew, and keep it up to date. I started looking at alternate DE's and installed Mate. In Unity changing between the languages works either by pressing Alt + Shift or by clicking the input method icon and choosing which language I need. In Mate it recognizes that I have Hebrew installed but does not change even the icon when I click Alt + Shift, and when pressing the icon and changing to Hebrew, the icon changes but nothing else does. I can read Hebrew online just fine even in Mate. What is the problem, and how do I fix it?
Don't know if MATE is supposed to be different in this respect, but on Unity this terminal command can be used to set input method:
gsettings set org.gnome.desktop.input-sources current 0
The number 0 (zero) means that you set the first available input method. Change it to 1 to set the second etc.
Maybe worth a try on your MATE desktop.
Installed MATE out of curiosity, and saw that it is different. Added an additional keyboard layout via System -> Preferences -> Hardware -> Keyboard -> Layouts, and encountered the same problem first. However, after having played around for a while, I could successfully switch keyboard layout. So maybe just relogin or reboot...