I have a personal site that has a totally different blogging audience than my business website; however, some content (like my About page) is almost exactly the same on both websites. Would I incur any type of penalties from search engines for having duplicate content if both sites have a link to each other?
As John Pointed out, the best way to intimate Google about duplicate content would be via a canonical tag. If not done, I am not sure if there will be a penalty but Google will certainly prefer one of the pages over the other.
In case you prefer the page from the business website to be picked by search engines, your canonical should be present on the personal website's page and should look like this
<link rel="canonical" href="<your business website's about page url>" />
A suggestion: If I were in your position, I will do this - I will elaborate on my business on the business about page with a little introduction about myself. I will place a link to my personal about page from there. Similarly, on the personal website's about page, I will elaborate on my career, track record, achievements, etc., with a brief introduction to my business. I will place a link to my business about page from there. By doing this I would have improved the content quality, relevance, and uniqueness of both pages. Plus I would have eliminated the need to place a canonical.