Depends on the person, but it's permanent for the most part. This is straight from the mouth of multiple therapists I've seen when I was younger. Confront the anxiety (like social anxiety), and so long as you're not an idiot (like say, hiding inside for 5 months right after), you'll be 95% fine after, no real time limit. Sure, I'll get spikes of anxiety, but they're maybe a thousandth of what I dealt with, a skipped heart beat, or a flutter of "oh man I used to puss out on this all the time", nothing more. Honestly, with a large majority of fears/anxiety, the only way to handle it is to confront it. For me personally, after doing something that slaps me in the face with the fear, I get over it, and I rationalize that it's not a logical threat, so I stop being scared. If I don't stop being completely scared, I turn my pansy ass around and confront it again until I have complete control, then if possible, push myself farther. For example, skydiving, I closed my eyes in the last 20 seconds, to ensure the WORST possible fear has happened, and I was still alive. Of course, that takes self-control, understanding anxiety and how it manifests, being driven to fix the issue, so I can understand for someone just blindly walking in, not knowing anything or being really motivated, it'd be a little harder.

One mistake a lot of people do is immediately run back to safety after confronting anxiety, which defeats the whole purpose. Don't take my word for it, take Mr. PHD's.

When facing a fear, it is critical not to engage in any safety behaviors that might “undo” learning; this is sometimes referred to as response prevention.

This is how you get the "I confronted it but I'm still scared" issue.

https://www.verywellmind.com/breaking-the-anxiety-cycle-1392987

(This is a great little overview of how and why it DOES work, just need to do it correctly)

That being said, it's entirely possible to completely overwrite things like anxiety, baseless fears, etc. That's literally how you get a bunch of dudes with swords to run at another bunch of dudes with swords. If you can teach people to run towards death, you can teach them to not be anxious, because I'd assume it takes a bit more work telling someone "Oh yeah, run towards the guy shooting at you, who just shot your friend, that's what you want to do", than say, something like social anxiety.

Again, this is for 99% of cases, I'm sure there's a few where something fucky is wired, it happens. But beleive me, I've seen some people really bitch, whine, hysterics, etc, but when there's an emergency, they don't even think about those things. It really greatly depends on how you confront the issue, and how you rationalize everything at the end. Of course if you go home telling yourself "oh man I almost died", you'll psyche yourself out again. Realizing that you are alive, therefore, your fear is incorrect, helps a lot, but that's an internalizing thing that I'm sure people do differently.

Either way, I've known plenty of people to overcome serious fears, myself included. It really comes down to wanting to get rid of that fear, and how you confront it.