High or low voltage on a split phase 240/120 VAC system is generally due to a loose connection, most often in the neutral or ground wire. If everything is in good condition, there is no reason your voltage should rise or fall on one leg, due to the load on the other.
If everything is sized correctly and all the connections are good and tight, you will see little or no dimming of lights when a large load is introduced (like an air conditioner starting up). Even then, the dimming will be temporary; motors draw large amounts of power when they start, which causes the dip in voltage. Once motors get up to speed (generally within one second), the voltage recovers and the lights return to normal brightness.
By tapping the center of the secondary coil, you have (in effect) two secondary coils, each with half as many turns in their windings as the entire secondary coil. This is why the voltage measured from either "leg" to neutral is 120V (the primary to secondary turns ratio has been doubled). Loading one half of that split secondary will have no effect on the voltage of the other half (since the turns ratio hasn't changed).
Below, is a link to a PowerPoint presentation for an explanation of overvoltage due to loss of neutral in a split phase system. Note that when the text states that the neutral is disconnected, the picture shows the ground being disconnected, so imagine that the center tap of the transformer has been disconnected as well.
http://code-elec.com/userimages/Lost Neutral.ppt
It is possible (and common) for a loose neutral connection to exhibit no problems until a large load is introduced.