  Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair
$\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal G$, we study the local
finite presentability and local coherence of the heart
$\mathcal{H}_{\mathbf{t}}$ of the associated Happel-Reiten-Smal{\o}
$t$-structure in the derived category $\mathrm{Der} (\mathcal{G})$. We start by
showing that, in this general setting, the torsion pair $\mathbf t$ is of
finite type, if and only if it is quasi-cotilting, if and only if it is
cosilting. We then proceed to study those $\mathbf t$ for which
$\mathcal{H}_{\mathbf{t}}$ is locally finitely presented, obtaining a complete
answer under some additional assumptions on the ground category $\mathcal{G}$,
which are general enough to include all locally coherent categories, all
categories of modules and several categories of quasi-coherent sheaves over
schemes. The third problem that we tackle is that of local coherence. In this
direction we characterize those torsion pairs $\mathbf t=(\mathcal T,\mathcal
F)$ in a locally finitely presented $\mathcal G$ for which
$\mathcal{H}_{\mathbf{t}}$ is locally coherent in two cases: when the tilted
t-structure in $\mathcal{H}_{\mathbf{t}}$ is assumed to restrict to finitely
presented objects, and when $\mathcal F$ is cogenerating. In the last part of
the paper we concentrate on the case when $\mathcal G$ is a category of modules
over a small preadditive category, giving several examples and obtaining very
neat (new) characterizations even in this more classical setting, also
underlying connections with the notion of an elementary cogenerator.
