Regarding solo air forks everything is clear. Once we bumped both chambers pressure is equal. But why for dual air forks air pressure doesn't try to get equal in both chambers? What's the point of second chamber if it has different pressure and doesn't try to equal it?
The negative air chamber provides resistance to the positive air chamber. It controls the speed air spring returns to its initial shape. The chambers are isolated so the air is unable to equalise.
The positive chamber provides resistance against compression, essentially how hard is the fork.
The third part of the suspension action is dampening, the negative air chamber controls the forks return and the damper stops if from oscillating and provides a smooth ride.
Rockshox explain dual air below quite well in suspension terms.
One huge advantage of dual air springs that isn't often mentioned, is that you can set sag independent of positive spring rate (air pressure). If you need to add a bunch of (+) pressure to resist bottoming, you can pump up the negative pressure to the point where you still get adequate sag. This means you can still have good small bump sensitivity at higher air pressures, where a solo air fork would stay topped out.
You will lose a bit of travel is negative pressure exceeds positive by more than a couple psi, but because the negative chamber is much smaller than the positive you can add a fair amount more negative without losing too much travel.
IMHO getting correct sag and spring rate is much more important than worrying about losing 5-10mm of travel from a higher negative pressure.
Note the opposite is true too, if you are getting too much sag at you desired running pressure (spring rate), you can reduce the negative pressure accordingly. This is valuable to people who like to setup their fork to use full travel rather than based on sag. If you aren't an aggressive rider but still wants full travel, then you will be running a low positive pressure, which will give you too much sag, so you can reduce your negative pressure until it sags between 15-25%.
Sidenote: Running excessive sag to achieve full travel is fair from ideal, you don't really gain any travel because whatever you gain at the end of the stroke you lose at the start from excessive (30-40%) sag. My 2009 Fox fork was like this, I found it best to run 15-20% sag and not worry about the fact I never used the last 3/4" of travel, the fork was way too wallowy if I lowered the pressure to use full travel. I replaced it with my dual air Revelation, and have been happy ever since. I can set my sag optimally and still enjoy every mm of travel it has to offer. Even though it only has an extra 10mm (130mm RS vs 120mm fox), it feels like an extra 2 inches.