  We derive and analyze the conditions for quantum conformal and Lorentz
invariance of the duality symmetric interacting chiral boson sigma-models,
which are conjectured to describe non-geometric string theory backgrounds. The
one-loop Weyl and Lorentz anomalies are computed for the general case using the
background field method. Subsequently, our results are applied to a class of
(on-shell) Lorentz invariant chiral boson models which are based on twisted
doubled tori. Our findings are in agreement with those expected from the
effective supergravity approach, thereby firmly establishing that the chiral
boson models under consideration provide the string worldsheet description of
N=4 gauged supergravities with electric gaugings. Furthermore, they demonstrate
that twisted doubled tori are indeed the doubled internal geometries underlying
a large class of non-geometric string compactifications. For compact gaugings
the associated chiral boson models are automatically conformal, a fact that is
explained by showing that they are actually chiral WZW models in disguise.
