It's hard to say without seeing your timeline and your design. If you want to duplicate a layer then select it in the timeline and go to Edit>Duplicate or press Ctrl/Cmnd + D
If the layer is a pre-comp then this will not duplicate the pre-comp so you can have separate animations in both, but you can change any other layer properties in the duplicate like position and rotation. If you need two different pre-comps then you duplicate the pre-comp in the Project Panel, open the duplicate pre-comp and change the internal animation there.
It is funny how one can sit for hours struggling with such a simple task. I am used to Ps where you can merely right click and select the duplicate option.
Luckily the two interfaces are similar, but I am only starting to learn Ae now.