I don't believe we've that much complex functionality supported in the app currently.
You can't copy/paste—as Sheena correctly stated—but you can move layers between compositions using CC Libraries.
To "copy" a layer, turn off the visibility of any unneeded layers by double-tapping their thumbnails, and then from the Share icon, select CC Library. You can then import the layer from that library into your other composition.
Hope that helps!
Here's a short (silent) video to show you how it's done.