You are correct: no information is lost in the Fourier Transform. What I meant is that it actually becomes harder representationally to compare localized amplitudes of the time domain information in the frequency domain. And typically, what you're looking at the frequency domain will tell you nothing about location data in the original frequency.

Here's why:

When you do the Fourier Transform (or in this case, the Discrete Fourier Transform (DFT)) you turn your time domain function into a frequency domain function. This function resides on the complex plane, with a real and imaginary part.

Typically, you're doing the transform to get at the information in the magnitude, which represents the frequency information in the original time domain function. To get at the magnitude, you do: sqrt(Re^2 + Im^2).

Now, when you're looking at the magnitude, you have lost location information entirely about the original signal. Rather, you now have perfect frequency information. (yes, combined with the phase: atan(Re/Im), you can reconstruct the original signal).

Here's an example. Imagine a time domain function x(n) where x(0) = 1, and x(n) = 0 everywhere else. This is just a spike at 1.

The magnitude of the DFT of that function is f(n) = 1 everywhere! In the time domain, you have total localization of the signal all in one point. But, in the frequency domain you have the opposite (constant 1 everywhere)! Intuitively, the reason for this is because in order for sines and cosines to represent such a localized function, you have to add up a lot of them in order to cancel each other out and produce such a localized signal.

tl;dr:

When you're comparing localized values in the time domain, the FT will most likely not be the tool to use. The FT gives you frequency information, without spatial information unless you take into account the phase. In this instagram puzzle example, the information about the pixel values at the edges of the strips have now been "spread" across many frequency values in the freq domain.

EDIT:

Another example is x(1) = 1 and x(n) = 0 elsewhere. This is just a time shifted version of the function given above. The magnitude of the DFT is still just a constant 1 everywhere. No difference! Only the phase differs.