There are quite a few different ways to wash your brushes. People use everything from bar soap to dish detergent to baby shampoo. Generally speaking, anything is better than nothing, but there are some pros and cons to every method.
Using soap meant for your skin or dirty dishes can dry out bristles. It’s not going to ruin your brushes, but it’s not great for natural fibers. If you want something cheap and easy, go for baby shampoo. It’s gentle, but effective.
If you need something a little more heavy duty for eyeliner and brow brushes, grab some olive oil. It’s total overkill for most brushes, but it’s exactly what you need for sticky gel formulas. Of course, this will be a waterless method so just dab a drop of oil on your brush then gently swirl it over a cloth or paper towel. Keep going until the brush stops leaving a mark.