Very interesting comments so far. No one mentioned the problem that a new professor may leave for a better position or may not be kept on for various reasons.
Unfortunately, this can happen with senior faculty too! Probably a bit less likely, but I've seen this happen numerous times
By the way, when a well established professor has major funding, how much of that does he or she keep for his or her own pay? I was not aware, until recently, that professors supplement their regular income from these funds.
This depends on a huge number of factors.
Traditionally, salaries in psychology departments are 100% effort for 9 months of the year. Meaning, grants or no, you get paid $x for 9 months of work. Typically, if you have your own grant you can pay to "continue" your salary over the summer.
Medical schools, hospitals, and other similar settings have a different arrangement. They may cover anywhere from 0-100% of your salary depending on the institution, the rest is up to you to get from grants. Faculty might be PI on 1 or 2 big R01s and cover the vast majority of their salary from that. They might be statistician on 20 different studies and get 5% of their salary from each.
There is a cap on the highest salary that NIH will pay (I believe in the neighborhood of $200,000), so in other words, if you are at 50% effort on a big grant and you have found some magical academic position the rest of us are jealous of that pays $500,000 a year, only $100,000 could be covered by the grant. That's based off my understanding of the system...we have some people much more senior than me here who might tell me I'm completely wrong
In addition to this, you can sometimes be hired as a consultant for a set $ amount, outside your salary (though I think you can also be a "consultant" and have it cover part of your salary). I think there are some limitations on this, when it can happen, how much it can be, etc. but I couldn't tell you offhand.
Again, I'm just a lowly grad student, but this is my understanding of the process. Those who have more experience with the process, let me know if I'm completely off base in anything I said.