Are professors' salaries a function of the amount of grant funding they obtain?

In the United States, the amount of money that a professor makes is not generally a percentage of the amount of funding that they have under management. Instead, professors are typically guaranteed a 9-month salary and can supplement with 3 additional grant-funded months (generally at the same rate) for summer. That salary is set by a combination of regulation, negotiation, and academic rank, in a manner vaguely similar to salaries at a large corporation.

Thus, high funding correlates with high salary because a well-funded professor is also likely to be progressing in their compensation package, but there is typically an effective "cap" on the amount that a professor can receive through their university (consulting, startups, and patent licensing fees may be an entirely different matter, but again are not actually determined by research funding, just correlated as being another common product of a successful research).

As for the actual compensation of specific faculty: for most US state-funded institutions, you can find complete salary databases published as public records. For example, here is one for Berkeley. Private institutions have relatively similar compensation rates (or a bit higher), since top private institutions are competing with top public institutions for the same talent.

I am just interested here in compensation for grant-making ability. So, if that is a factor in salary, I would like to know the effect on salary.

While it is very unusual, the University of Arkansas has a Faculty Salary Funding Incentive Plan faculty can earn cash rewards up to 25% of their regular salary for bringing in grant money that covers their regular salary.

But, the total amount of this "bonus" is still a percentage of the base salary, not a percentage of the grant funding.

In the United States, two major sources of grant funding are the NSF and the NIH, both of which limit how much PI salary may be charged to grants:

• The NSF limits the amount of salary senior personnel can receive from grant funding to 2/9 of their regular base salary. (See this question for more on that.) This roughly corresponds to a professor getting up to 2 months of their base salary from grant funding, while they typically get 9 months from the university.
• The NIH limits the salary that one may receive from an NIH grant with a salary cap.

Professors work hard to get grant funding not because they earn a lot of money from it, but because (a) it's often expected for promotion and tenure, and (b) it allows them to buy equipment and hire students to help carry out the research they want to do.

In Germany, one Euro in grants will typically not automatically increase your salary by x Euro-Cents. Instead, professors will have (possibly multi-year) targets: "Over the next three years, bring in at least x Euro grant money as principal investigator." If you reach your target, you will get a temporary salary increase (Leistungszulage) until the next target review. Reaching your target multiple times in a row may lead to this increase becoming permanent. (Of course, professors' targets usually also include publications, teaching and service.)

As elsewhere, more success in acquiring grant money will strengthen your position on the job market. You will get more offers and have a better bargaining position for Berufungszulagen and Bleibezulagen, so grant success correlates with your salary indirectly.

I have attempted to describe the German academic salary system here: Do professors in Germany have other payment than their standard salary?