bubbers44,
the resulting control input from both sticks is just the simple algebraic sum of both sticks' deflections. If the F/O makes a control input of, say, 7 degrees to the left (call it "-7"), and the Captain makes a 12 degree input to the right ("+12"), then the resulting control input is -7 + 12 = +5.
If the F/O had been using full left input, the best the Captain could achieve without using the takeover priority button would be neutral, by applying full right deflection.
There is a known incident in which the Takeover button was used successfully in an emergency situation. The Captain's sidestick was connected in such a way that the roll input was reversed. The F/O noticed the problem just in time, pressed the takeover button, and landed the aircraft safely. See also the entry
Lufthansa A320, incorrectly-wired sidestick, Frankfurt am Main, March 2001 in our
Compendium CRICA.
There are also cases where cross-controlling without using the takeover button occurred. I assume it's a matter of training, and the fact that this happens is the strongest argument against the non-backfed, uncoupled sidesticks. For this to develop into a real problem usually also involves a serious breakdown of CRM.
The takeover button also serves to disconnect the autopilot, e. g. when taking over for the manual landing after an autopilot approach, so it may be a well-trained thing to do when starting to move the stick. (One can also disconnect the A/P by overcoming a restraining force that is keeping the sidestick "lcoked" with A/P active.)
Bernd