Yes, they do. The supposed advantage of a twin engine plane is that you can fly if one quits. The FAA makes you demonstrate that you can handle the plane on one engine before they turn you loose with a twin-engined plane, but the unfortunate reality is that the fatal accident rate for twin engined (small, piston) planes is worse than singles. The problem is that the probability of an engine failure is twice as high in a twin as in a single. In a single, if the engine quits, you're a glider, and the plane is fairly straightforward to handle. As long as you don't hit anything too solid, you'll be OK if you keep the plane under control through the landing. In a twin though, you have to do everything just right, or you can end up out of control and hit the ground nose down and die.

In most light twin planes, both props turn the same direction, and so the plane has a natural left-turning tendency when climbing. If the left engine goes out, you're in a very bad situation, because now the plane really wants to go to the left. If you indulge it, you don't get to go where you want. If you don't, you may get a worse climb rate (which is already very bad.)