You cannot use the “Duplicate” or “Duplicate Special”-tool to duplicate along with a curve. We are going to use an animation-tool to get a similar effect.
First, create your object and the curve.
Before you duplicate your object, you need to decide how many objects you need. Then you can set the Animation Range from Frame 1 to Frame (2* the count of objects that you need.)
Select your object and the curve. Now use Constrain > Motion Paths > Attach to Motion Path. When you play the animation you will see that your object moves along the curve.
We need to change the keyframe tangents to linear to distribute the objects evenly over the entire curve. To do this select your object again and open the Graph Editor (Windows > Animation > Graph Editor). Then select both keys and select “Linear Tangents”.
To create all our objects, select the object and “Visualize > Animation Snapshot (options)” In the options, you need to set End Frame to your Animation Range Last Frame and the Increment to 2.
Delete the original object and curve.
While this technique is a little cumbersome, it is very flexible and extensible. For example, you could add a scale animation prior to creating the snapshot, or add deformers to modify your object prior to duplicating it.