When I click multiple times in a row on “Estimate score”, without changing anything on the board, the score and confidence is different every time. It isn’t a small difference, for example I just had the following scores for the same game: 34, 76, 44, 31, …
If it is beginning of game, (first 10-30 moves), it is normal for estimator not to be sure about score. If you click multiple times, it has to calculate again and again, but with only few stones placed, there is a lot of possibilities, alternative moves, and so much free space, so it is very hard to predict what is going to happen. After every few moves, it starts to stabilize the result. Just like your groups get their shapes.
Estimate score is far from buggy. Do not underestimate it, if you don’t know how complicated that system is. 
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The site uses an approach that randomly simulates many possible play outs, which leads to a random (and often different) estimate each time. It leads to a very rough estimate that one shouldn’t put too much faith into, especially when the game is far from being over.
Estimating the score for an ongoing game is difficult since one can’t predict exactly how the players will actually play it out (e.g., mistakes, securing the win rather than expanding the lead, etc.). Even calculating a score estimate from determing the theoretically perfect play from the current position will only result in a guess of how the game will really turn out.
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All good. The estimate score IS NOT buggy ! 
Seriously, thanks for your clear answer and description. I hope this will help other players wondering what’s going on.
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I wonder if it’s worth revisiting this feature. I find it an very unreliable indicator of score, even in the latter stages and at the very end of some games. One instance had me losing by 20+ points on score estimates in the last move, only for me to win. If you compare http://gyazo.com/979dc173a212a171748714c86419cef9 and http://gyazo.com/83c44635f58eadbe74560be70a2fb64d you can see a current game of mine in the endgame. I just ran estimate score twice and got black to win by 25 and 48 points - there surely isn’t that much uncertainty in this game…?
I wonder if it would be worth enhancing the functionality a bit so that you can not only select stones that are definitely dead, but also ones you think are definitely NOT dead. That should make it possible for the player to customise the estimate to his/her own assessment of the most likely territorial outcome. Views anyone?
The score estimator is just that, an estimator, and in both of your screenshots it also tells you how much it confides in its estimation:
You will even notice that if you click the score estimator three times, it may give you three different estimates (with according %ages), and if you click it four times, well, you get it, I assume 
Also, given how many possibilities there still are in the pictured boards, I think, so many things can still happen.
I believe that one of the reasons we don’t yet have programs that play with 9p strength is due to the fact that a proper estimation of the score during a running game is everything but trivial. I believe this is also one of the reasons why we enjoy the game more than Tic Tac Toe 
One could probably run an instance of Zen for every running game and have it watch the game closely so that it can better estimate the score but I fear we’d all have to pay in gold for server costs then. Not to speak of licensing …
You can simply enable the estimator in the analysis mode. The estimator is not just an estimator when the game is close and within 30 seconds the player needs to make a choice whether to make a defensive move and lose one point, or play a more aggresive one more to keep up: the choice depends on the estimate and it must be precise (which can be done by helping the estimator with a few dummy moves in the analysis move).
People should learn how to count the score in their head. That will lead to real progress. If they rely on the estimator to decide whether to defend or not then they will never improve beyond a certain point until they stop using the estimator.
It is actually a part of the game and skill to be able to count the score.
I watched all the Haylee’s recordings on youtube: she easily defeats 8d players on Tygem (people say she is 3p in Korea), but even she doesn’t have time to count and many times she said something like “in fast games with 30s per move I dont have time to count” and does that only if her opponent spends too much time on a move. And, btw, the same can be said about the analysis mode itself: it’s nice to be able not to use it, but it’s there and it’s useful in certain cases.
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