The MATE desktop; when used with dual monitors by default only shows the panels on one screen or the other. The solution is to right-click on the panels and opt to add a new panel. Then drag this new panel to the other screen end on to the first panel. This will give you a panel on each screen. Then move your applets to the new panel by right-clicking on them and moving them. Then you can have what looks like one long panel spanning both monitors. This is the only way to do this that I could find. I cannot get this to work on the bottom panels as well, but having a Macintosh styled top panel is pretty cool though. So give this a try if you want to have a desktop with the proper panels spanning behaviour. The Xfce4 desktop can have proper panel spanning behaviour but this is the MATE desktop we are talking about. I hope they fix this bug soon, this is annoying when you want both screens to look the same.
The picture below shows the end-result of this trick. This is the best that you can do when you want to have a panel on each screen. The only other way is to have one panel on the top of one screen and one on the bottom of the other, but this is a better way to accomplish this.