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(I've pushed the old answer down)

I think (what currently is) Lemma 5.9 (tag 05VM) of the Algebraic Spaces chapter of the stacks project is exactly what we want.

Let F,G fppf sheaves and let $F \to G$ be a schematic, flat, locally of finite presentation and surjective (as in: surjective on fields). Then it is an epimorphism of sheaves.

I would be very interested in the converse...

EDIT: old answer.

I guess we can sum it up as follows (after a-fortiori's comments).

In general there is no chance of having sections everywhere: $\mathbf{A}^1 \setminus 0 \amalg 0 \to \mathbf{A}^1$. And other types of counterexamples can be cooked up by throwing in infinite residue field extensions or non-separability.

For smooth morphisms however we always get sections! (by EGA IV-3 17.16.3)

For $f: X \to Y$ a flat morphism of affine schemes we have $f$ faithfully flat if and only if $f$ is surjective. If $f$ now is a smooth morphism of schemes we can use the aforementioned result in EGA and get $etale$-local sections. I would guess that for algebraic spaces we also get $smooth$-local sections by pulling back to an atlas, and the same should hold for algebraic stacks.
show/hide this revision's text 2 added 420 characters in body

(I've pushed the old answer down)

I think (what currently is) Lemma 5.9 (tag 05VM) of the Algebraic Spaces chapter of the stacks project is exactly what we want.

Let F,G fppf sheaves and let $F \to G$ be a schematic, flat, locally of finite presentation and surjective (as in: surjective on fields). Then it is an epimorphism of sheaves.

I would be very interested in the converse...

EDIT: old answer.

I guess we can sum it up as follows (after a-fortiori's comments).

In general there is no chance of having sections everywhere: $\mathbf{A}^1 \setminus 0 \amalg 0 \to \mathbf{A}^1$. And other types of counterexamples can be cooked up by throwing in infinite residue field extensions or non-separability.

For smooth morphisms however we always get sections! (by EGA IV-3 17.16.3)

For $f: X \to Y$ a flat morphism of affine schemes we have $f$ faithfully flat if and only if $f$ is surjective. If $f$ now is a smooth morphism of schemes we can use the aforementioned result in EGA and get $etale$-local sections. I would guess that for algebraic spaces we also get $smooth$-local sections by pulling back to an atlas, and the same should hold for algebraic stacks.
show/hide this revision's text 1 [made Community Wiki]

I guess we can sum it up as follows (after a-fortiori's comments).

In general there is no chance of having sections everywhere: $\mathbf{A}^1 \setminus 0 \amalg 0 \to \mathbf{A}^1$. And other types of counterexamples can be cooked up by throwing in infinite residue field extensions or non-separability.

For smooth morphisms however we always get sections! (by EGA IV-3 17.16.3)

For $f: X \to Y$ a flat morphism of affine schemes we have $f$ faithfully flat if and only if $f$ is surjective. If $f$ now is a smooth morphism of schemes we can use the aforementioned result in EGA and get $etale$-local sections. I would guess that for algebraic spaces we also get $smooth$-local sections by pulling back to an atlas, and the same should hold for algebraic stacks.