As previously suggested, you can use the context of the ItemsRequesting client-side event, in order to pass any parameters to the specified WebServer. However, regarding your scenario, I would suggest you a slightly different approach.
When some DropDownList is expanded, its request to the WebService is triggered automatically. This is why, I would suggest you to handle the aforementioned client-side event - OnClientItemsRequesting and pass only the ID of the DropDownList, which triggered the event. In such a manner, you can access this ID in the underlying WebService and perform the data binding, regarding the control which triggered the event.
I have prepared a sample project for you, demonstrating the implementation of the above suggestion. In the clientItemRequesting, you can observe how the ID (custom parameter) is passed to through the context and in the WebService - how it is accessed.
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