Professional mathematicians are not always professors. There are many working for government-based research labs, other government agencies, private research firms, and industry.
In these cases, in my experience many of them -- because they do not teach as a regular activity -- do not retain many of the details, but they understand all the concepts.
Many of these folk might not remember things like trigonometric integrals, or they might not be able to recall the proper way to solve certain classes of ordinary differential equations, but they certainly understand the math -- they just might not be able to recall it off-the-cuff.
When you work in industry or government, you have ample access to resources. For instance (just to use an off-the-cuff example), if I see $\int \arctan x\ dx$, I don't know offhand what that is. But I know it's something, and I know how to find it, and I can fully understand all of the details on how to compute it. So if it comes up in my work, I can recognize that and go from there. Memorizing it isn't particularly useful for me. I haven't had to compute the antiderivative in 15 years. But if I needed to know how to do it, it would take me less than 5 minutes to figure out.