My current airplane has two engines, but is not guaranteed to accelerate to rotation speed nor climb on one engine. So if either engine fails during the take-off run, my actions are the same as those of the single engine pilot: stop. I retard both throttles to idle and brake as heavily as required. If there's any risk of going off the runway I'll secure the engines, if I can, but maximum braking and directional control are going to take precedence over the single pilot reaching I'd have to do to get to the emergency fuel shutoff valves and the magneto switches. I might settle for just chopping the mixtures. That's when a second pilot would come in handy. And SOPs are to ensure we wouldn't just be in each other's way as we headed for the emergency exit.
In general, the slowest speed you can fly an airplane is its stall speed. Below that speed the wings won't produce enough lift to counteract the airplane's weight, and the airplane will drop instead of flying. Stall speed increases with weight and with bank angle, which is one of the reasons why turning around can be a risky manoever.
Assuming I get into the air, the lowest safe speed for me to fly is not the stall speed (although a stall poses the same dangers to me as to the deliberately single-engine pilot), but a higher speed, the minimum controllable airspeed, written Vmc. It's sometimes marked on the airspeed indicator with a red line. If I experience an engine failure and allow the airspeed to decay below that red line I may not have sufficient control authority to counteract the turning moment of the remaining engine. The airplane could roll over on its back, from which position flight becomes undesirable. Hmm ... while theoretically it might be possible to fly a light twin inverted on one engine, I'm going to advance the position that this mode of flight is untenable. Video clips of airshow performers demonstrating me wrong won't change my desire never to be there.
My normal rotation speed is above my Vmc, but trying to climb after loss of an engine would result in rapid loss of airspeed, so it would be possible to take off, start to climb, lose an engine and be have my speed decay to Vmc before I could react. Here is where the SOPs differ among various companies. The engine failure at take-off briefing might start "In the event of a failure before rotation ..." or it might be "In the event of a failure below Vmc..." In either case the briefed action will be to stop or land straight ahead. Even if the runway is insufficient and the terrain is inhospitable, it's safer to land with some control than to continue to fly at a speed at which the airplane is uncontrollable.
When my airplane reaches rotation speed I raise the nose, wait for the main wheels to come off, verify climb and insufficient runway remaining, then select gear up. I try to regard reaching to raise the gear as reaching a decision point, and not as a routine post-takeoff action so that I'm always keyed and thinking about the possibility and the actions I need to take if the engine fails. If I've already selected gear up, and the engine fails, I've made my decision to fly. I control the yaw with rudder (and a tiny bit of bank), advance full power on the remaining engine, and maintain exactly blueline speed. A blue line is drawn on the airspeed indicator at the best single engine rate of climb speed.
The failed engine at this point would be sitting there worse than useless, because instead of powering forward its side of the airplane, it's holding it back. I can't jettison it, but I can effectively hack off the propeller, turning the blades perpendicular to the airflow. This is called feathering. The trick is that the propeller needs to be still going around for feathering to work, so it has to be done quickly,
That's the reason for the saying that in many twin-engined airplanes, the purpose of the second engine is to carry you to the site of the crash when the first one fails. In an engine failure situation, it's vitally important to control direction and airspeed. Engine failure on take-off accidents are frequently the result of the pilot losing control of one of those factors. I'm not saying that that's easy, or even always possible, considering the obstacles that may be present, but you don't need to be a pilot to recognize the importance of keeping the airplane right side up and going in a safe direction.
So I have takeoff power set, brakes released, airspeed alive, gauges green, max power confirmed, rotate, nose up on the attitude indicator, positive rate of climb shown on both the altimeter and VSI. With insufficient runway remaining, I raise the gear. That's when I'm committing myself. An engine failure right there with the gear cycling would be bad.
If I am airborne but cannot maintain flight sufficient to avoid obstacles and manoever back to the runway, then I revert to the same situation as the pilot of the single. I choose the best spot within gliding distance to put down the airplane. If the terrain is unsuitable, I have to choose whether to do it gear up (extended wheels could catch and cause the airplane to flip over) or down (the gear tearing off helps to dissipate the energy of the crash and protect the cabin from objects on the ground).
So at any point in my roll, I have a plan for what to if an engine failure occurs. That way I don't have to make split second decisions. You see that there is still a point at which even perfect reflexes can't protect me from ending up on my belly in the weeds. That's why I may elect not to take off at night from a particular runway, or refuse a full load. When deciding where I can take off and what I can carry I must always consider single engine performance.