  We consider random trigonometric polynomials of the form \[
f_n(x,y)=\sum_{1\le k,l \le n} a_{k,l} \cos(kx) \cos(ly), \] where the entries
$(a_{k,l})_{k,l\ge 1}$ are i.i.d. random variables that are centered with unit
variance. We investigate the length $\ell_K(f_n)$ of the nodal set $Z_K(f_n)$
of the zeros of $f_n$ that belong to a compact set $K \subset \mathbb R^2$. We
first establish a local universality result, namely we prove that, as $n$ goes
to infinity, the sequence of random variables $n\, \ell_{K/n}(f_n)$ converges
in distribution to a universal limit which does not depend on the particular
law of the entries. We then show that at a macroscopic scale, the expectation
of $\ell_{[0,\pi]^2}(f_n)/n$ also converges to an universal limit. Our approach
provides two main byproducts: (i) a general result regarding the continuity of
the volume of the nodal sets with respect to $C^1$-convergence which refines
previous findings of Rusakov et al., Iksanov et al. and Aza\"is et al., and
(ii) a new strategy for proving small ball estimates in random trigonometric
models, providing in turn uniform local controls of the nodal volumes.
