I clogged my sink up with pumpkin seeds last Halloween.
On the outside wall of your house exactly behind the sink there might be a trap (unless you live in an apartment building or a newer home). If there is take a wrench out there. Open. Flush. Close. See what happens.
Also you can always put a bucket beneath the elbow joint beneath the sink and remove it. There will probably be a lot of food stuck in there. Before you do so, though, I highly suggest you bail out the water from both sinks, or else you're gonna make a big mess.
If you're not comfortable with minor plumbing call a rooter service, come and snake the pipe, but charge you from $50-$100 bucks.