Unfortunately, there's no way to copy and paste animations at this time. This is something we're looking to add in a future release, but in the meantime there's a way to work around this.
If you have a layer that has animations you want to use on other layers, the "Duplicate Layer" button is your best friend.
Select the layer with animations you want to duplicate and click the "Duplicate Layer" button at the top of the layer list.
This will copy the layer and all of the applied animations and interactions. You can then use the Appearance property of the layer, to change the asset being used.
While this isn't a graceful solution, it does work when you need to have similar animations on more than one layer.