As an Aikido beginner, this is how I have come to view the use of Atemi in my dojo:
Beginners are not being taught atemi in general because they are learning to focus on their technique. The basics of balance, footwork and good ukemi are maybe viewed as the important first step.
If Atemi were simply another motion integral to the technique then I'm sure this would not be the case. Instead the use of Atemi seems to be a more thoughtful matter that involves how it affects Uke during the technique that's being performed.
I occasionally [rarely] have achieved a clear state of mind while performing a technique which allowed me to be aware of the activity around me. This is something that more skilled people take for granted and is an important part of making sure that you do not throw your Uke into another person or a wall. I believe that when my skill has advanced to a point that this awareness is common, I will be more prepared to evaluate the properly timed use of Atemi during the execution of a technique.
In preparation we are being taught to be aware of what both hands are doing throughout the technique. There are many times that one hand is simply "out there" in a blocking position. These are possible opportunities for Atemi if the result would be to refocus my Uke's attention aware from the direction that I am moving.
When should it be used? It appears to me that there are points at which you would like to distract your uke from the fact that you are performing a ballet around one of his arms. A quick strike to the face while moving into my Uke would certainly give him something to think about other than the fact that his arm has just been moved 2 feet from his body. There also appear to be moments when the next step I take may open my body up to a strike from a thoughtful Uke. Perhaps the proper Atemi would keep him from noticing the unprotected part of my body and allow him to direct an attack there.
How much power should be directed into an Atemi? I don't know, I'm just a beginner, but I do know that if someone slaps me in the face my mind will suddenly be focused on that point. It seems to me that the power of the strike only needs to be strong enough to get Uke's attention. I don't need to practice for hours on a heavy bag because I'm not attempting to devastate Uke with the power of my attack, it's not a karate punch where the strike IS the technique. All I want to do is say, "Look at the hand! Look at the hand!" while I twist him into a pretzel.
The proper timing and type of Atemi should be chosen based upon the affect that it has on Uke. A magician would not distract your eye TOWARDS his other hand which is slipping a ball under the cup. As such Atemi is a slight of hand that says to Uke, "hey, the fun is over here...in your face," while really I am preparing him to bow deep and kiss the floor. It also has the power to trick Uke into commiting his energy in a direction that is advantageous to my impending purpose.
This is my perspective as a beginner. It is all together likely that I will change my view as I grow more skilled but I believe that my current view point will always have value to me as an objective first perspective on this technique.