We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let $X_1, \ldots, X_n$ be independent random variables with common distribution the symmetric Jacobi measure $dμ(x) = C_d (1-x^2)^{\frac d2 -1} dx$ with dimension $ d \geq 1$ on $[-1, 1]$, and let $μ_n = \frac{1}{n} \sum_{i = 1}^{n} δ_{X_i}$ be the associated empirical measure. We show that $\lim_{n \to \infty} n\E \left[ W_2^2( μ^n, μ) \right] = \sum_{k = 1}^{\infty} \frac{1}{k(k+d-1)}$, where $W_2$ is the quadratic Kantorovich distance with respect to the intrinsic cost $ρ(x, y) = |\arccos(x) - \arccos (y)|$, $(x, y) \in [-1, 1]^2$, associated to the model. When $μ$ is the product measure of two Jacobi measures with dimensions $d$ and $d'$ respectively, then $\E \left[ W_2^2( μ^n, μ) \right] \approx \frac{\log n}{n}$. In the particular case $d = d' = 1$ (corresponding to the product of arcsine laws), $\lim_{n \to \infty} \frac{n}{\log n} \E \left[ W_2^2( μ^n, μ) \right] = \fracπ{4}$. Similar results do hold for non-symmetric Jacobi distributions. The proofs are based on the recent PDE and mass transportation approach developed by L.~Ambrosio, F.~Stra and D.~Trevisan.