It appears to actually be the combination of the plot title and the sgtitle. If you just have one or the other, it works fine. My recommendation would be to add the sgtitle first, then the subplots. This way, when you add the annotation, the axes are in their new position.
I did notice that I had to include a drawnow to force the update before getting the axes position.
figure()
sgtitle('Scenario 1: Test Title')
for c = 1:3
subplot(1,3,c)
scatter(randi(5,[1,5])-3,randi(9,[1,5])-5)
axis([-2 2 -4 4])
xlabel('x-axis')
ylabel('y-axis')
title("P_" + num2str(c))
drawnow
p = get(gca, 'Position');
anno = annotation('arrow');
anno.HeadStyle = 'plain';
set(anno, 'X', [p(1) p(1)], 'Y', [p(2) p(2)+p(4)]);
anno2 = annotation('arrow');
anno2.HeadStyle = 'plain';
set(anno2, 'X', [p(1) p(1)+p(3)], 'Y', [p(2) p(2)]);
end
