Tom's solution is a great one. It allows you to work with the transition as is.
One other possibility is to take the matter into your own hands and use Motion to adapt the transition to work the way you want - that is, assuming you have Motion (which you should) and are willing to try a bit. Don't worry, it is nondestructive. That is where the "Open a COPY in Motion" is very handy.
Each transition is different.
For example, take the "Light Noise" transition: it merely plays a little quicktime movie over your whole frame.
If you open a copy in Motion, you can apply image masks so that the the "light noise" is only visible where the text is.
In the example below, I applied two image masks to the "light-effect" movie, set In and Out points so that the Mask A is applied in the first part (while clip A is visible), and Mask B in the second. Take the source for Mask A to be Transition A (which is the Motion placeholder for the outgoing clip) and the source channel to be Alpha (transparency)), and likewise for B. Works like a charm.
