I'm currently finding this to be a pain in the butt right now as well. So I would love to see what people suggest.
Take your spriter object and choose a different entity.
Assuming you have your player1/2 as different entities.
@lucid recommended me to choose the object from the layout and you can choose the starting entity for each instance there.
But then you also need to keep in mind they are the same object. So you have to keep track of instances.
If your project has more animations and entities this only becomes increasingly confusing and difficult to work with and get animations, instances, and entities to all play nicely together.
My current issue is how to work with instances not created in the editor/at runtime. Instances created dynamically make this object much harder to work with than your standard objects.
In my case this one object is my:
So I am currently at a loss how I am supposed to manage my spriter project in construct at this point.