Venting problem?

I have installed a double sink in my kitchen with a garbage disposal under the left sink. My problem is this, When we operate the garbage disposal what ever is left in the pipe work blows back up into the right sink causes a mess and winds up getting the clean dishes full of garbage, and the water that comes up really stinks.
How can I correct this? I have added a "P" trap to the sink on the right but still have this problem. Any help would be appreciated.
Sincerely Pete

Venting Problem, Most Likely Not...

...But it does sound like there is a cloggged kitchen drain line that needs to be snaked and cleaned by a professional wiith the right equipment. Food stuff is more than likely the cause of the stoppage.
When the drain is free flowing again, don't forget use only cold water and let it run after the grinding stops for several seconds.

Thanks to all. I'm going to try both suggestions