To remove game score as a factor, Vic Ferrari wrote an app at his website timeonice.com to look at even-strength play with the score tied. I ran the numbers for each team in the league during the 2007-08 season. The full table is below, but here are a couple of comments:
Correlation between number of shots faced and save percentage: .097. Correlation between shot differential and save percentage: .063. Based on this sample, outshooting appears to have little effect on save percentage at even strength in a tie game.
Correlation between blocked shots and save percentage: -0.239. I'd like to see more data on this, but it seems to suggest there could be a weak negative relationship between blocked shots and save percentages (i.e. it is tougher for a goalie to play on a team that blocks more shots).
These results are more sensitive to luck/randomness since the sample size is smaller. There appears to be a clear shot quality effect, however, since most of the elite defensive teams show up near the top with weaker teams near the bottom. There is also a gap of .041 between the best and worst teams.
I compared the teams' results with Alan Ryder's shot quality numbers. Most teams had similar rankings, but there were a few teams with large discrepancies. Either those teams had very good or very bad goaltending, or they were very good or bad at protecting leads or playing from behind, or there was something else going on.
Better than expected:
Anaheim: 29th in shot quality, 1st in ES SV% while tied
N.Y. Rangers: 25th in shot quality, 6th in ES SV% while tied
Dallas: 27th in shot quality, 17th in ES SV% while tied
Montreal: 17th in shot quality, 8th in ES SV% while tied
Worse than expected:
Columbus: 1st in shot quality, 20th in ES SV% while tied
Washington: 4th in shot quality, 22nd in ES SV% while tied
Buffalo: 13th in shot quality, 30th in ES SV% while tied
Calgary: 3rd in shot quality, 18th in ES SV% while tied
Tampa Bay: 14th in shot quality, 29th in ES SV% while tied
Anaheim and Buffalo are the most interesting ones here. I don't think that either Giguere is that good or Miller is that bad. It could be that both teams have extreme results in the opposite direction when they are leading or trailing. However, I suspect that for both of those teams shot quality simply may not accurately measure the true difficulty of scoring chances against.
Assuming the RTSS shot distance figures that make up the shot quality rating are more or less correct (and these are road numbers only, to reduce rink reporting bias), the difference would therefore have to be either because of the average shot angle, the average defensive pressure on the shooter, or the quality of the average opposing shooter. Behind the Net has a shot quality measure that takes into account shot angle. According to those numbers, Giguere faced shots of about average difficulty, while Miller faced slightly easier than average shots against. It looks like the Ducks probably kept more of their shots away from the middle of the ice than normal, but that doesn't explain the entire difference.
I would suspect that as an offensive team Buffalo might be giving up more rush chances and more space to opposing forwards, which allows them more time to evaluate their options and make their shots. Anaheim's defence-first philosophy likely creates more pressured shots, which are easier to stop. Another possible explanation is that Anaheim has an elite checking line which may help their goalies by reducing shots against from the opposition's best players.
I would love to see these numbers for when teams were leading and trailing as well, to see both the overall save percentage effects and which teams were great at locking it down or opening it up.
Here are the full numbers for 2007-08 at even-strength, game tied: