Regarding checking instruments - this needs to be treated with extreme caution.
In most light twins, during the take-off and initial climb it is likely the engines are generating their max power anyway, so it is unlikely a further increase could occur. Possible, such as a prop runaway, but unlikely.
The instruments on a dead engine can seem very live. What's the MP pressure on an engine providing no power? Somewhere around 29-30" depending on the conditions. Unless feathered, a prop could still be providing 22-2300 rpm while it windmills. These two indications can make the engine look like it is providing power during the heat of battle.
A PA31 incident at Archerfield many moons ago proves this. The pilot believed he was still generating power based on what he saw on the instruments and elected not to secure the failed engine. It took him around 12nm to climb to 1000 odd feet, but hey, he must have been generating power right? Subsequent investigation found he saw close to what I spoke about before.
While an increase in power can lead to misidentification of a failed engine, one needs to pick the time and place to start feeding this info to fresh, overwhelmed minds.
In any case, moving the throttle through its travel, and checking there was no change in yaw or noise, should allow you to identify whether there is a problem and avoid securing a perfectly good engine.