  Let $E/\mathbb{Q}$ be an elliptic curve with full rational 2-torsion. As d
varies over squarefree integers, we study the behaviour of the quadratic twists
$E_d$ over a fixed quadratic extension $K/\mathbb{Q}$. We prove that for 100%
of twists the dimension of the 2-Selmer group over K is given by an explicit
local formula, and use this to show that this dimension follows an
Erd\H{o}s--Kac type distribution. This is in stark contrast to the distribution
of the dimension of the corresponding 2-Selmer groups over $\mathbb{Q}$, and
this discrepancy allows us to determine the distribution of the 2-torsion in
the Shafarevich--Tate groups of the $E_d$ over K also.
  As a consequence of our methods we prove that, for 100% of twists d, the
action of $\operatorname{Gal}(K/\mathbb{Q})$ on the 2-Selmer group of $E_d$
over K is trivial, and the Mordell--Weil group $E_d(K)$ splits integrally as a
direct sum of its invariants and anti-invariants. On the other hand, we give
examples of thin families of quadratic twists in which a positive proportion of
the 2-Selmer groups over K have non-trivial
$\operatorname{Gal}(K/\mathbb{Q})$-action, illustrating that the previous
results are genuinely statistical phenomena.
