After reading the other answers here and pondering them, I decided to throw my own answer in anyway and flesh it out a little. I am posting this after having accepted another answer.
How much knowledge is (minimally) necessary?
I don't think my initial instincts are that far off from what others are saying, though perhaps the emphasis is from another angle (hence this is worth posting). It still comes back to being able to do the day-to-day bookkeeping oneself. The level of knowledge minimally required to do this with standard accounting packages on the market is pretty low. It amounts to "generally understanding some terminology and being willing to learn how to use software aimed at beginners." It's a very low bar today. Whether or not you hire a bookkeeper you need to know enough to be able to supervise what the bookkeeper does, since the bookkeeper is probably handling your money.
Being able to do the bookkeeping means you can do some very basic things yourself. You can print invoices, find out who owes you money, pay your expenses, write your own checks, and balance your checkbook. It does not mean you have to understand the finer points of day count conventions, determine what needs to be adjusted end of year, or worry about exactly when income is realized.
The second thing one needs to be able to do is formulate questions to ask an accountant. So there you have it. If you can run accounting software (or are willing to learn), ask questions, and understand the answers, that's sufficient.
However, accounting is power for business owners
If you are going to be self-employed with multiple streams of revenue for any period of time, more accounting knowledge is better. What accounting knowledge gives you, fundamentally, is an ability to analyse your revenue stream when you are making business decisions. Now if you are just independent contracting (one customer at a time, each for an extended period), you can probably figure such things out without any accounting knowledge at all, but if you are consulting for multiple customers concurrently, and offering different services, then such knowledge is quite empowering when it comes to actually making business decisions. Being willing to learn a little at a time as you need it is thus a very good thing.