It depends on which restaurant you go to. At Pinnochio's Village Haus and the counter at Typhoon Lagoon we were told that the fries that come with the hamburger are already the side and we could not order the side.
On the other hand, Beaches and Cream does not count the fries and they allow you to order any side that you want (the waiter even allowed us to order the Chicken Caesar Salad as a side (I got filled up on just that).
Unfortunately, you have to ask these employees ahead of time (and they don't always look too thrilled about having to answer your questions - mostly at the counter service places in the parks).
Many times, at the Counter-type places, the vouchers are not a good deal at all (you may even lose money), since you can't order another side + the dessert is sometimes just a cookie.
If you go to mousesavers.com they tell you which restaurants are worth it and I think they only included 3 restaurants that they feel are worth the $11 for the voucher.
We did find that Pecos Bill's Cafe was well worth the voucher (although mousesavers.com does not list it as being a good deal).