Assume (1). Choose a morphism $\mathcal{X} \to X$ into an algebraic space $X$ which turns $\mathcal{X}$ into a gerbe over $X$. Let $X' \to X$ be a surjective, flat, locally finitely presented morphism and set $\mathcal{X}' = X' \times_X \mathcal{X}$. Note that $\mathcal{X}'$ is a gerbe over $X'$ by Lemma \ref{lemma-base-change-gerbe}. Then both squares in $$ \xymatrix{ \mathcal{I}_{\mathcal{X}'} \ar[r] \ar[d] & \mathcal{X}' \ar[r] \ar[d] & X' \ar[d] \\ \mathcal{I}_\mathcal{X} \ar[r] & \mathcal{X} \ar[r] & X } $$ are fibre product squares, see Lemma \ref{lemma-cartesian-square-inertia}. Hence to prove $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is flat and locally of finite presentation it suffices to do so after such a base change by Lemmas \ref{lemma-descent-flat} and \ref{lemma-descent-finite-presentation}. Thus we can apply Lemma \ref{lemma-local-structure-gerbe} to assume that $\mathcal{X} = [U/G]$. By Lemma \ref{lemma-gerbe-with-section} we see $G$ is flat and locally of finite presentation over $U$ and that $x : U \to [U/G]$ is surjective, flat, and locally of finite presentation. Moreover, the pullback of $\mathcal{I}_\mathcal{X}$ by $x$ is $G$ and we conclude that (2) holds by descent again, i.e., by Lemmas \ref{lemma-descent-flat} and \ref{lemma-descent-finite-presentation}. \medskip\noindent Conversely, assume (2). Choose a smooth presentation $\mathcal{X} = [U/R]$, see Algebraic Stacks, Section \ref{algebraic-section-stack-to-presentation}. Denote $G \to U$ the stabilizer group algebraic space of the groupoid $(U, R, s, t, c, e, i)$, see Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-stabilizer-groupoid}. By Lemma \ref{lemma-presentation-inertia} we see that $G \to U$ is flat and locally of finite presentation as a base change of $\mathcal{I}_\mathcal{X} \to \mathcal{X}$, see Lemmas \ref{lemma-base-change-flat} and \ref{lemma-base-change-finite-presentation}. Consider the following action $$ a : G \times_{U, t} R \to R, \quad (g, r) \mapsto c(g, r) $$ of $G$ on $R$. This action is free on $T$-valued points for any scheme $T$ as $R$ is a groupoid. Hence $R' = R/G$ is an algebraic space and the quotient morphism $\pi : R \to R'$ is surjective, flat, and locally of finite presentation by Bootstrap, Lemma \ref{bootstrap-lemma-quotient-free-action}. The projections $s, t : R \to U$ are $G$-invariant, hence we obtain morphisms $s' , t' : R' \to U$ such that $s = s' \circ \pi$ and $t = t' \circ \pi$. Since $s, t : R \to U$ are flat and locally of finite presentation we conclude that $s', t'$ are flat and locally of finite presentation, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-flat-permanence} and Descent on Spaces, Lemma \ref{spaces-descent-lemma-locally-finite-presentation-fppf-local-source}. Consider the morphism $$ j' = (t', s') : R' \longrightarrow U \times U. $$ We claim this is a monomorphism. Namely, suppose that $T$ is a scheme and that $a, b : T \to R'$ are morphisms which have the same image in $U \times U$. By definition of the quotient $R' = R/G$ there exists an fppf covering $\{h_j : T_j \to T\}$ such that $a \circ h_j = \pi \circ a_j$ and $b \circ h_j = \pi \circ b_j$ for some morphisms $a_j, b_j : T_j \to R$. Since $a_j, b_j$ have the same image in $U \times U$ we see that $g_j = c(a_j, i(b_j))$ is a $T_j$-valued point of $G$ such that $c(g_j, b_j) = a_j$. In other words, $a_j$ and $b_j$ have the same image in $R'$ and the claim is proved. Since $j : R \to U \times U$ is a pre-equivalence relation (see Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-groupoid-pre-equivalence}) and $R \to R'$ is surjective (as a map of sheaves) we see that $j' : R' \to U \times U$ is an equivalence relation. Hence Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap} shows that $X = U/R'$ is an algebraic space. Finally, we claim that the morphism $$ \mathcal{X} = [U/R] \longrightarrow X = U/R' $$ turns $\mathcal{X}$ into a gerbe over $X$. This follows from Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-when-gerbe} as $R \to R'$ is surjective, flat, and locally of finite presentation (if needed use Bootstrap, Lemma \ref{bootstrap-lemma-surjective-flat-locally-finite-presentation} to see this implies the required hypothesis).