One of my yoga teachers has us find the one that is currently dominant in the cycle before we practice nadi shodhana pranayama, and then select the starting nostril based on that.
I wonder if I'm unusual in that I can consciously control which nostril is open or closed. Most of the time the switching from side to side is unconscious but if I concentrate, I can cause it to happen consciously. This is really useful when I have a cold and a blocked nostril and I want to get some relief when blowing my nose doesn't help - I just focus on allowing that side of my sinuses to open and after a few minutes it will happen.
I'm probably one of them then, because I've tested this on and off for years whenever someone makes the claim, and it's never true for me.
It's just really annoying how many people say with absolute certainty that everyone does it.
If you do not have this behaviour (which can disappear with certain neurological diseases or just idiopathically not exist) you also likely have a reduced sense of smell.
There is a little bit of airflow in the constricted nostril, but nowhere near comparable to the other. You can test this by covering up your nostrils individually and comparing how easy it is to breathe through them. Then wait a half hour and try it again. (Note that a minority of people do not have this.)
However, this is not something you can consciously control or override, any more than you can override contractions in your intestine.