Unlike Arizona's CCW laws (which provide certain benefits to permit holders, even with Constitutional carry), there is no benefit in Alaska to have a permit besides reciprocity. Most Alaskan's who travel "down south" to the "lower 48" for pleasure go to either Hawaii or Washington State (I'm not kidding with this), neither of which honor Alaska's permit. I was going to get mine when I lived in Alaska because I traveled to Arizona a lot, but then they went Constitutional Carry, and it became moot.
I'd imagine the number of permit holders in Alaska is pretty small, and probably not much more than 1 or 2% of the population (just over 700,000 or so). I never knew anyone who actually received a permit after they got Constitutional Carry, and only knew of a couple prior to that.