How can I implement the ability to 'cut' sprites into arbitrary shapes?

I want to create a game where you have scissors that cut a sprite in a shape that you create by dragging your finger on the screen.

My problem is that I can’t figure out how can I cut my sprite in the specific shape that the user creates with their finger.

So far the only solution that I found searching the web is that I’ll need to create my own shader that lets me make a specific part of a texture transparent, but don’t understand how I can apply that to my scenario?

I would suggest using a texture on a mesh, instead of a sprite. That way you can remove those parts of the mesh the user cuts. You could also cut it into two meshes.

Not rendering at all is cheaper then rendering transparent.