Yes, you can definitely do this with drivers. Here's how:

In this first image you can see that I've created 2 spheres, each with a Basis shapekey. Sphere 1 has a second shapekey called Sphere1Key, which changes the sphere to a fatter shape. This key is animated (keyframed) from 0 to 1 over 50 frames. You can see the value field is green due to the keyframed status.

The second sphere also has a second shapekey called Sphere2Key, which deforms the sphere in the same way.

For the Sphere2Key shapekey, I have right clicked on the value field and selected Add Driver.

With Sphere 2 still selected I open up the Graph Editor and switch from F-Curve Drivers mode with the dropmenu in the toolbar.

You can see in the left panel that the Sphere.001 is selected and I've also selected the Value for the Sphere2Key shapekey.

Now we just have to map the value from the first sphere's shapekey to this one.

To do this, press N to bring up the right property panel. In the Drivers area we want these settings:

1. Change type to Scripted Expression.
2. In Expr, add a variable name (I've used getSphere1KeyValue)
3. Click Add Variable button
4. In the variable name field paste in the name from the Expr box above (getSphere1KeyValue)
5. Change the variable type to Single Property
6. Change the Prop type to Key
7. Change the selected Key id block to "Key" or whatever the name of the id block for the keyframed data for the Sphere1 shapekey value is (The available values list automatically when you click in this field. If you're not sure what it should be you can select Sphere1 and the switch to the Dope Sheet and find it there).
8. Enter the RNA path to the sphere 1 key value into the Path field. You can get this value by right clicking on the Sphere1Key field in the shapekeys panel, and choosing Copy Data Path.
9. The value for the Value field in the Driver panel should now update when you scroll the timeline back and forth.
10. If the Sphere2 shapekey doesn't respond, you may need to click the Update Dependencies button in the driver panel. You should then see the Sphere2 shapekey copy the Sphere1 shapekey keyframed values in synchronization.

Yes, you can definitely do this with drivers. Here's how:

In this first image you can see that I've created 2 spheres, each with a Basis shapekey. Sphere 1 has a second shapekey called Sphere1Key, which changes the sphere to a fatter shape. This key is animated (keyframed) from 0 to 1 over 50 frames. You can see the value field is green due to the keyframed status.

The second sphere also has a second shapekey called Sphere2Key, which deforms the sphere in the same way.

For the Sphere2Key shapekey, I have right clicked on the value field and selected Add Driver.

With Sphere 2 still selected I open up the Graph Editor and switch from F-Curve Drivers mode with the dropmenu in the toolbar.

You can see in the left panel that the Sphere.001 is selected and I've also selected the Value for the Sphere2Key shapekey.

Now we just have to map the value from the first sphere's shapekey to this one.

To do this, press N to bring up the right property panel. In the Drivers area we want these settings:

1. Change type to Scripted Expression.
2. In Expr, add a variable name (I've used getSphere1KeyValue)
3. Click Add Variable button
4. In the variable name field paste in the name from the Expr box above (getSphere1KeyValue)
5. Change the variable type to Single Property
6. Change the Prop type to Key
7. Change the selected Key id block to "Key" or whatever the name of the id block for the keyframed data for the Sphere1 shapekey value is (The available values list automatically when you click in this field. If you're not sure what it should be you can select Sphere1 and the switch to the Dope Sheet and find it there).
8. Enter the RNA path to the sphere 1 key value into the Path field. You can get this value by right clicking on the Sphere1Key field in the shapekeys panel, and choosing Copy Data Path.
9. The value for the Value field in the Driver panel should now update when you scroll the timeline back and forth.
10. If the Sphere2 shapekey doesn't respond, you may need to click the Update Dependencies button in the driver panel. You should then see the Sphere2 shapekey copy the Sphere1 shapekey keyframed values in synchronization.

Note: you may notice in the gif that the numbers in the Value field are not changing while the animation runs. This is simply a UI display issue, the actual values are changing, as evidenced by the shape change in Sphere.001. If you hover the cursor over those fields you'll see them update.