Ok first things first. Just to make sure we're on the same page here:
You won't see any secondary button, you only see a smooth shell on the Mighty Mouse surface. The sensors are under the plastic shell:
As dtravis pointed out earlier in this thread you need to configure the buttons by going to the
System Preferences ->
Keyboard & Mouse ->
Mouse tab. By clicking on the drop-down menus you can select which action you want the button to have. In Apple's jargon
Primary Button means left-click,
Secondary button means right-click.
A member here found out that the sensors (image I posted just above) are ultra sensitive so you need to
raise your fingers off the mouse when you right-click otherwise the sensors think you are actually left-clicking. It's not a question of how much pressure you apply for the click. it's if the sensors feel a weight, however slight, on the left side, the mouse registers a left-click. So by raising your fingers off the mouse, only keeping the one that taps on right side, it will work unless it is a defective unit.
I hope I am being clear here.
So if you can't get anything working with the mouse, I would suggest you take it back to an Apple Store and get a replacement.