Copy and paste while in the same web tab does work. You could create a model that is only used for storing some components, then go and copy any you need for the new model, and paste them in.
You also could download SKP files of the components you might frequently need, and then insert them straight from your desktop into the web model.
Those things are possible with the Free version too. The reason to have Shop is if you need any of the extra features, or if you are using SketchUp for paid work.