Does everyone need a set in a LCG
Tl;dr: One set probably suffices for Limited play, but you probably need more than one to play Constructed unless you want to limit what people can build.

Most LCGs (and ECGs, which are the same thing), offer two broad categories of play: Constructed and Limited. 

In Constructed play, you build decks from the cards available, usually with some restrictions like a maximum number of copies for each card. ECGs I'm familiar with come with at most enough cards for a single player to build an arbitrary Constructed deck (sometimes many fewer, e.g. in Epic Card Game you are allowed up to 3 copies of each card but the set comes with only 1 of each). So to play Constructed, you'll probably need multiple sets, or you'll need to (potentially severely) restrict what each player can build. For example with one copy of Warhammer: Invasion and two players, you could decide that one player will build an Order deck, one will build a Destruction deck, and no one will use more than half the copies of any neutral card; with four players you could just assign each player a race.

In Limited play, you build a deck immediately before playing, often using a random and/or competitive building process. For example, there's often:


Sealed - deal each person a random pile from the communal pool, then let them remove cards they don't want until they're down to minimum deck size.
Draft - deal each person cards from the communal pool, then each selects a card and passes the rest to another person. Keep selecting and passing until everyone has a deck (or more than a deck, and then cuts some out).


There are many variants of these, and the ECGs I'm familiar with all support some kind of Limited play using only one set. For example, in Epic Card Game you can draft with up to 4 people out of one set.