I'm currently using the following equipment for my street photography.
I'm using an Olympus OM-D E-M5 Mark II mirrorless camera. It's a small, inconspicuous camera that's perfect for street photography. It gives you all the functionality of a DSLR, but in a very small package. The camera has a nice retro design too. I like it a lot.
I primarily use prime lenses when shooting street photography. The lens I use most often is the Olympus 17mm F/1.8 prime lens. Occasionally I put on an Olympus 45mm F/1.8 lens, when I want to make street portraits.
I do have an Olympus 12-50mm F/3.5-6.3 lens, but I don't use this lens very often for street photography. I don't like having to zoom in or out. With a fixed focal length, I know where I have to stand to get the framing right. That way I can take the shot immediately, without having to waste valuable time zooming and seeing the scene disappear before my eyes.
The Olympus OM-D E-M5 Mark II is a great camera, but I'm not that impressed with its low light performance. Therefore I like to use flash when needed. It also makes the colors pop!
I use a Metz Mecablitz 52 AF1 Digital Flash unit. It's pretty powerful and has all the functions I need and it was reasonably priced.
Using flash in street photography can be difficult, because you don't have time to set up your flash unit like in a studio. To avoid harsh shadows and blown out faces, I use a Gary Fong Lightsphere Collapsible. As you can see in the photos above, it produces a very even lighting.
The downside of using a big flash unit with a Lightsphere on top, is that you're not that inconspicuous anymore :-(