Have you had a look at any of the examples in the documentation? E.g. https://docs.pyvista.org/api/core/_autosummary/pyvista.dataobjectfilters.scale

scale returns a copy by default. You need to either assign the output to a variable or use inplace=True

Thanks for your reply. I just tried that too. I used both inplace=False (with another variable) and True (on the same variable) and tried different scale values. 'mesh2' appears exactly as 'mesh1'. Of course, since I saved them both as .ply and the files are identical.
I really wonder how all these operations can do absolute nothing, w/o any error or warning indicated ..

A more detailed response:

I used sphere2 = sphere.scale(0.5) as you suggested. Same result. The only difference now is that the saved PLYs are not exactly the same. Yet, their plots appear the same.
About pl.set_scale() you are of course right. So I did it correctly, wishing that at least this works. Alas!
What a stubbornness! 🙂

Now, I have a question: Have you tested any of these yourself? If so, did they work? Which of them exactly did work with you?

How can I physically scale the objects/meshes directly? (On an individual basis and independently of other objects present, of course.)

This code already given does that. I have expanded it with the camera position.

import pyvista as pv

pl = pv.Plotter()

sphere = pv.Sphere(center=(0, 0, 0))
cone = pv.Cone(center=(0, 1, 0))

sphere.scale((0.5, 0.5, 0.5), inplace=True) # Resize sphere to half
cone.scale((2.0, 2.0, 2.0), inplace=True)   # Resize cone to double

pl.add_mesh(sphere)
pl.add_mesh(cone)

# add in a fixed camera position to keep it from resizing the view automatically, if desired
# pl.camera_position = cpos
pl.show()

[Edited: fixed the example which had a mistake from copying and pasting two different codes together.]

Where is 'cpos' obtained from? Can you please also add it in your code?
E.g., if I set it at start, with cpos = pl.camera_position and apply it just before plotting or with plotting (pl.show(cpos=cpos)) then I get a huge sphere that covers everything:

See the end of this section https://docs.pyvista.org/user-guide/simple.html#plotting. There are a few different methods to get the camera position after interaction, one of which is given in that example. You can also use something like camera_position = pl.show(return_cpos=True).

I know that. I already mentioned in my last reply.
Never mind. I will close this discussion.
Thank you for your time, @MatthewFlamm!