As everyone who read the two threads (1 in Kuvasz Discussions and 1 in Great Pyrenees) is aware, I was very concerned about Sapphire's fearfulness last week, although I probably overreacted at first.
However, I thought you might like an update on how Sapphire is doing after one week.
I have been walking her every evening; there were only two days when almost no one was out. I was unable to walk Sapphire on two of the mornings, but usually no one is out in the mornings anyway. She sometimes completely panic, usually because she is afraid of someone or has seen a trash can being moved or a bicycle being ridden. Her tail gets curled way between her legs, her ears are down and flat against her head, and she just RUNS. There is some improvement here: she is doing this less and less, but she still gets scared. Now, rather than run, she often starts going fast, but not running, toward home; she then pulls against the leash, and if I am already going in that direction, she pulls me along faster than I would like to go. Except when she is scared like this, she doesn't pull, so I think of the pulling as a fear issue rather than a leash training issue. Whenever she sees someone, she gives an alert/slightly nervous look: ears up but not fully forward, tail down. This is distinct from her very alert/agressive look: eyes fully up and fully forward, tail held almost straight up and often waving it also. Sometimes she barks at these people; sometimes she does nothing. When the person approaches, however, Sapphire often backs away and barks, which I correct by a tug to prevent her pulling on the leash or sheltering behind me. On a better note, she will allow anyone, once I get her to stand still, to hold out their hand to her; if she doesn't like them she will back away and sometimes bark. If they persist and attempt to pet her (I allowed one man who seemed to understand the danger posed by a fearful dog to attempt petting her after she backed away from him in this manner) she gives a warning growl and would probably snap. Otherwise, they can pet her. She has adjusted to most of the other things she sees on walks, except cars that pass nearby. She wants to run in the opposite direction, but I would rather it stays that way.

The last thing I want is for Sapphire, should she ever be out in the road, to think that running cars are okay.
The aggressive attitude I mentioned never comes except when someone comes into our house and she is upset and barking/growling. After she realizes that we don't mind, which generally takes several minutes even with sharp leash corrections, she occasionally givers her alert look (without any nervousness in it) and barks some more. So this seems to be improving, although not as fast as I would have liked.
Overall, this seems to be the case. There is some improvement, but it is slow and will take lots of time and effort to solve. So, I'm learning that training dogs takes time, especially when the dog doesn't have a very people-oriented, really-wants-to-please-you attitude.
As an aside, she has an ENTIRELY different attitude at Petsmart. The class we took previously did help, but it was not in the way we had hoped.

We went there this Saturday for the socialization, even though the class was cancelled for the next two weeks and we will have to cancel our enrollment in it. Once she realized where we were, she hurried in to find the trainer and enthusiastical
ly greeted everyone we saw at first with her friendly attitude. Her ears were way back, so that you couldn't see them from the front, whole rear end wagged along with her tail, and she rushed toward people, trying to sniff them. Once she got over that, she acted a little more wary of people, but is still far more cooperative with people, tolerating all of them unless they approach her. There are a few people that she doesn't like and barks at when they come towards us, but she is fine with almost everyone. She is also less nervous around dogs there, but she is quite wary enough to get away from those that growl, snap, or otherwise act aggressive. The difference is so big it would absolutely hilarious if it weren't for the disappointment of her behavior in our neighborhood. I still think it is a little funny.

I will probably post pictures of Sapphire this weekend.
Sofia