You'll need to clone the objects you want to change as a separate object, then place the new one on your second layout.
To remove the original object from your second layout only and not the first, delete it from the layout view, not the object/project bar on the right.
Also you can filter your objects bar to only show objects that exist on the current layout by right clicking - filter objects.
Objects existing globally across all your layouts can be very useful, but in your case if you want your layouts to have different objects, you should actually create different objects. You can make the new ones based on the old ones by cloning objects.
Here are the relevant manual entries: