  According to recent progresses in the finite size scaling theory of
disordered systems, thermodynamic observables are not self-averaging at
critical points when the disorder is relevant in the Harris criterion sense.
This lack of self-averageness at criticality is directly related to the
distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of
samples $(i)$ of size $L$. In this paper, we apply this analysis to disordered
Poland-Scheraga models with different loop exponents $c$,corresponding to
marginal and relevant disorder. In all cases, we numerically obtain a Gaussian
histogram of pseudo-critical temperatures $T_c(i,L)$ with mean $T_c^{av}(L)$
and width $\Delta T_c(L)$. For the marginal case $c=1.5$ corresponding to
two-dimensional wetting, both the width $\Delta T_c(L)$ and the shift
$[T_c(\infty)-T_c^{av}(L)]$ decay as $L^{-1/2}$, so the exponent is unchanged
($\nu_{random}=2=\nu_{pure}$) but disorder is relevant and leads to non
self-averaging at criticality. For relevant disorder $c=1.75$, the width
$\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ decay with the same
new exponent $L^{-1/\nu_{random}}$ (where $\nu_{random} \sim 2.7 > 2 >
\nu_{pure}$) and there is again no self-averaging at criticality. Finally for
the value $c=2.15$, of interest in the context of DNA denaturation, the
transition is first-order in the pure case. In the presence of disorder, the
width $\Delta T_c(L) \sim L^{-1/2}$ dominates over the shift
$[T_c(\infty)-T_c^{av}(L)] \sim L^{-1}$, i.e. there are two correlation length
exponents $\nu=2$ and $\tilde \nu=1$ that govern respectively the
averaged/typical loop distribution.
