Yes, it is possible to use WhatsApp on both SIMs on a dual SIM smartphone. What you need to do is to run a separate account of WhatsApp which is different from your original WhatsApp account. You can do this with a modified version of the official WhatsApp. This modified version is known as OGWhatsApp (so named apparently after the name of the person who created it).
To get the OGWhat's App, though, you need to first uninstall the regular WhatsApp. Of course, remember to save all the data of WhatsApp before uninstalling it! For a complete step by step guide refer: http://webtrickz.com/using-both-whatsapp-numbers-on-android-dual-sim/