No. Effect controls are only drawn in the context of the currently active effect and that's no different for the Puppet Tool.
I was watching a tutorial on pre-composition in After Effects and was wondering how to merge the two layers together so I could see all the puppet pins at ones?
There is nothing keeping you from copying a layer in one comp and pasting it in another. There is also no reason that you can't have multiple layers with puppet pins in the same composition. That's the normal way to work.
The other option is to set up edit this look at that by dragging your timelines apart and stacking them and adding another comp viewer:
That's a bunch of keyframes for something that should be as simple as just animating a leg moving. It looks like you used the puppet pin recording feature. Check your other post for some ideas.