I am sorry but I am lost. If they both have probabilities or amplitudes of 1 would that not lead to a joint probability of 200% in the nonidentical case and 400% in the identical one?
The statement that protons are indistinguishable is not strictly correct either, because protons have a spin. Protons with the same spin are indistinguishable, but you can tell apart protons with different spin. The spin of protons can only assume two values, so effectively there are two classes of protons, indistinguishable within the class.
In your specific case, it is clearly false that the probability of having one particle in one place is 200%. However, my statement still holds for expectations, and you end up with an expected two particles in one place. In the indistinguishable case, you must compute expectations based on amplitudes, not probabilities.
Going back to the experiment that I described, you can imagine that the particles are released at A and B with opposite spins, and then the detector at A’ only detects the spin that corresponds to the particle at A. This causes you to measure yet another probability, distinct from the other two, because there are now more possibilities and there are still multiple ways to cause the detector to find something. It could detect the proton from A, but the proton from A could also have its spin flipped and thus not be detected. The particle from B could arrive at A’ with the wrong spin and not be counted, or it could have its spin flipped along the way and be counted. You still cannot tell which proton you detected!
Similar complications occur with polarization of photons, which someone else mentioned in one of the comments. It’s worse though because polarization is a continuous quantity, and there are more ways to change it.
Incidentally, amplitudes are actually complex numbers. You can think of them as little arrows, like this: →, or this: ↖. In fact, these arrows are also rotating with the passage of time; they trace out little circles. To calculate the probability, we square the absolute value of the complex number. The absolute value of a complex number is equal to the length of the arrow, and squaring a length gives you an area. Thus the probability is essentially the same as the area of the circle traced out by the rotating arrow.
Events with high probability correspond to long arrows (big amplitudes), and low probability events have short arrows (small amplitudes). Amplitudes can cancel out when added together if they point in opposite directions. Thus we observe that some sequences of events have very low probability. We sometimes say that these events interfere with each other.
Sometimes this interference seems mysterious, as in the double–slit experiment, and other times it seems very mundane. In real life we rarely bother to calculate the probability that the batter will hit the ball before the pitcher throws it, but calculating the correct answers in quantum mechanics requires taking into account many such unusual events.