Does a curse need to hit a person in order for it to work?

Depends on the Curse

As you noted in your question, it does seem like most curses (and spells in general) hit only one person at a time as usual. While nothing definitive can be said on the matter (since there is no instance of any knowledgeable character like Dumbledore saying X spell or X class of spell hits only one/multiple people, and JK hasn't weighed in to my knowledge) there are a couple inferences that can be made.

1: Most curses are single-target spells. Things like the Killing Curse and the Stunning Spell are depicted as emitting "jets" or "bolts" of light, and if the light hits a person/thing that person/thing suffers the effect. That's pretty open and shut.

2: Other spells, like Expelliarmus and the Impediment Jinx (not a curse to be sure, but relevant here I think) can target one OR MORE people. And it does seem to be an effect that can be "cast" across a general area. For example, Harry uses Impedimenta in Harry Potter and the Goblet of Fire in the graveyard and hits "multiple" death eaters, even though he's casting the spell wildly over his shoulder. Your example with Neville is likewise an "imprecise" casting.

So what's going on? I think for certain curses, anything caught in the arc of the want as the spell is cast is getting hit. We know Expelliarmus at least has bespoke wand movements involved in the casting. A precise cast, with sharp quick movements, creates a smaller "Cone" of effect which hits only one person. Meanwhile casting the same spell with sweeping want movements (which may be easier to get the "pattern" right) projects over a wider area. Which would explain why Longbottom hit multiple people with his inaccurate spell.

The question now becomes "Why not always cast such spells as wide as possible to ensure you hit your target in a fight?" The answer I think is twofold, one practical, one theoretical. Practically speaking, using the wand movements to create a larger effect means moving your wand over a larger distance, thereby taking longer to cast the spell. In a duel with competent enemies (think Dumbledore v Voldemort in Harry Potter and the Order of the Phoenix or Snape vs the staff in Harry Potter and the Deathly Hallows fractions of a second matter. So the quicker you launch your spell the better! Theoretically too (though I hasten to add we have no PROOF of this) "spreading out" your curse in such a manner may make it easier for individuals to block. Shield charms in particular seem to be castable at different "power levels" and some curses/jinxs are described as "well-aimed" which may be an indication that keeping your curse "tight" makes it more powerful/harder to block than a wider-cast spell.

I realize this answer is in large part conjecture, but as there does not seem to be a firm in-universe or straight-from-JK reason for what we're seeing/reading, conjecture is all we have!