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The scale-invariant glitch statistics observed in individual pulsars (exponential waiting-time and power-law size distributions) are consistent with a critical self-organization process, wherein superfluid vortices pin metastably in macroscopic domains and unpin collectively via nearest-neighbor avalanches. Macroscopic inhomogeneity emerges naturally if pinning occurs at crustal faults. If, instead, pinning occurs at lattice sites and defects, which are macroscopically homogeneous, we show that an alternative, noncritical self-organization process operates, termed coherent noise, wherein the global Magnus force acts uniformly on vortices trapped in a range of pinning potentials and undergoing thermal creep. It is found that vortices again unpin collectively, but not via nearest-neighbor avalanches, and that, counterintuitively, the resulting glitch sizes are scale invariant, in accord with observational data. A mean-field analytic theory of the coherent noise process, supported by Monte-Carlo simulations, yields a power-law size distribution, between the smallest and largest glitch, with exponent $a$ in the range $-2\leq a \leq 0$. When the theory is fitted to data from the nine most active pulsars, including the two quasiperiodic glitchers PSR J0537$-$6910 and PSR J0835$-$4510, it directly constrains the distribution of pinning potentials in the star, leading to two conclusions: (i) the potentials are broadly distributed, with the mean comparable to the standard deviation; and (ii) the mean potential decreases with characteristic age. An observational test is proposed to discriminate between nearest-neighbor avalanches and coherent noise., 39 pages, 11 figures. Accepted for publication in ApJ