When I started shooting with my .22 (a rifle and later, a revolver), I collected my brass for the 1st year and a half. Below is a pic. Best I can tell, there are over 30k cases in that box.
I then started competing with my only other gun - a brand new .357mag S&W 686. After about 5 years, I bought another 686 and relegated the first one to back-up status with about 70k rounds though it.
I don't know if it takes 100k+ rounds for "the transformation", but "thousands" might be on the low end. "Hundreds" is a single range session for some.
While I do not disagree with the value of large quantities of practice, I think a lot depends on the discipline and the level of proficiency desired.
I've trained a number of shooters in rifle and pistol to the level of "Expert" in the NRA system, and many of them have gotten there with well under 1000 rounds fired in the discipline. These shooters have gone on to place well and even win matches at the local and state level well before they have fired 10,000+ rounds in their discipline. Of course, I've also worked with good people who simply did not have a knack for shooting and might not make "Expert" or win even a local match with 100,000 rounds of practice.
I suppose higher round counts are likely necessary to become competitive in the action pistol disciplines at the national level (and maybe even some of the more competitive local areas), but I do not have the breadth of experience to be sure. But since I have had rifle shooters post perfect scores (200-20x) in NRA events and also place first in national matches, I am confident that that level of proficiency can be acquired with well under 10,000 rounds.
In my own shooting experience, it only took a few hundred rounds to become proficient as a deer and varmint hunter, increasing my kill rate to well over 90% of shots fired. But densely wooded areas and smaller farm fields in E Texas don't afford many shot opportunities at over 200 yards, and deer, coyotes, etc. are pretty big targets at those ranges.
I've also had the chance to help a couple dozen people without much prior rifle experience kill their first (and sometimes second and third) deer. With care and 100 or so rounds of practice, they almost never miss with proper guidance on how to avoid the most common rookie mistakes. At that level of practice, my focus is on getting them to realize their limitations, identifying their expected level of accuracy with each shooting position, and pass on shots where they can't use a shooting position where they have proven a high level of accuracy.
But I do recommend the structure and organization of the NRA Marksmanship Qualification program to get shooters to higher levels of shooting proficiency more quickly and with less ammo cost that less structured plinking and simply sending rounds downrange. I see a lot of people at the range with so many holes in their target that it is difficult to see where each shot is hitting. Knowing the outcome of each shot is key to productive practice, as is a structured program for monitoring improvement and gradually increasing the level of challenge in an attainable manner. The NRA Marksmanship Qualification Program provides that in a number of disciplines.