For nearly everyone who might have altitude issues, after about 3 nights, they go away. If you have carried loads over passes in the Alps, this isn't much different. The air is only slightly thinner, so you go a little slower -- that's about it.
The heavy breathing and thin air is not the same as altitude sickness. The two most prevalent AMS symptoms are loss of appetite (to the point of nausea for some), and headache. Everyone struggles in the thin air, but that's not AMS.
