  We present a framework to study generic neutron-star binaries in
scalar-tensor theories of gravity. Our formalism achieves this goal by suitably
interfacing a post-Newtonian orbital evolution (described by a set of ordinary
differential equations) with a set of non-linear algebraic equations, which
provide a description of the scalar charge of each binary's component along the
evolution in terms of isolated-star data. We validate this semi-analytical
procedure by comparing its results to those of fully general-relativistic
simulations, and use it to investigate the behavior of binary systems in large
portions of the parameter space of scalar-tensor theories. This allows us to
shed further light on the phenomenon of "dynamical scalarization", which we
uncovered in [Barausse, Palenzuela, Ponce and Lehner, Phys. Rev. D 87,
081506(R) (2013)] and which takes place in tight binaries, even for stars that
have exactly zero scalar charge in isolation. We also employ our formalism to
study representative binary systems, obtain their gravitational-wave signals
and discuss the extent to which deviations from General Relativity can be
detected. The insights gained by this framework allow us to additionally show
that eccentric binaries can undergo scalarization/de-scalarization phenomena.
