How to get box-shadow on left & right sides only

NOTE: I suggest checking out @Hamish's answer below; it doesn't involve the imperfect "masking" in the solution described here.

You can get close with multiple box-shadows; one for each side

box-shadow: 12px 0 15px -4px rgba(31, 73, 125, 0.8), -12px 0 8px -4px rgba(31, 73, 125, 0.8);

http://jsfiddle.net/YJDdp/

Edit

Add 2 more box-shadows for the top and bottom up front to mask out the that bleeds through.

box-shadow: 0 9px 0px 0px white, 0 -9px 0px 0px white, 12px 0 15px -4px rgba(31, 73, 125, 0.8), -12px 0 15px -4px rgba(31, 73, 125, 0.8);

http://jsfiddle.net/LE6Lz/

I wasn't satisfied with the rounded top and bottom to the shadow present in Deefour's solution so created my own.

inset box-shadow creates a nice uniform shadow with the top and bottom cut off.

To use this effect on the sides of your element, create two pseudo elements :before and :after positioned absolutely on the sides of the original element.

div:before, div:after {
  content: " ";
  height: 100%;
  position: absolute;
  top: 0;
  width: 15px;
}
div:before {
  box-shadow: -15px 0 15px -15px inset;
  left: -15px;
}
div:after {
  box-shadow: 15px 0 15px -15px inset;
  right: -15px;
}

div {
  background: #EEEEEE;
  height: 100px;
  margin: 0 50px;
  width: 100px;
  position: relative;
}

<div></div>

Edit

Depending on your design, you may be able to use clip-path, as shown in @Luke's answer. However, note that in many cases this still results in the shadow tapering off at the top and bottom as you can see in this example:

div {
  width: 100px;
  height: 100px;
  background: #EEE;
  box-shadow: 0 0 15px 0px #000;
  clip-path: inset(0px -15px 0px -15px);
  position: relative;
  margin: 0 50px;
}

<div></div>

Negative spread and Masking shadow

CSS box-shadow uses 4 parameters: h-shadow, v-shadow, blur, spread:

box-shadow: 10px 0 8px -8px black;

The blur parameter adds the gradient effect, but adds also a little shadow on top and bottom borders. To get rid of this side effect we can use:

• Negative spread reduces the shadow on all borders: you can play with it trying to remove that little vertical shadow without affecting too much the one obn the sides (it's easier for small shadows, 5 to 10px.)

• Masking shadows of the same color of the background (white in this case), which allows for ticker shadows. Note that this masking shadow needs to have blur = 0 to fully cover the side effects.

Here two examples, the second one uses Masking shadow:

div{
    width: 100px;
    height: 100px;
    border: 1px solid green;
    margin: 10px;
    float: left;
}
#example1{
    box-shadow: -10px 0 8px -8px black, 10px 0 8px -8px black;
}
#example2{
    box-shadow: 
        0 -6px white,
        0 6px white, 
        -7px 0 4px -3px black, 
        7px 0 4px -3px black;
}

<div id="example1"></div>
<div id="example2"></div>

If none of these approaches suit your needs, you can also add an absolute div on the side of any existing divs.

Just remember to set the container div as position: relative so this absolute div will stay inside.

#example3 {
  position: relative;
  width: 100px;
  height: 100px;
  margin: 10px;
  border: 1px solid green;
}

.shadow {
  position: absolute;
  height: 100%;
  width: 4px;
  left: 0px;
  top: 0px;
  box-shadow: -4px 0 3px black;
}

<div id="example3">
  content here
  <div class="shadow"></div>
</div>

Try this, it's working for me:

    box-shadow: -5px 0 5px -5px #333, 5px 0 5px -5px #333;