If you want $k$ users to be able to reconstruct $s$ and no smaller number of users to be able to learn anything about the secret, you must have a polynomial $f$ of degree $k-1.$
The dealer gives exactly one share $(x_i,f(x_i))$ to user $i,$ for $i=1,2,\ldots,n$.
That is all that user gets. It doesn't really matter which user gets which share.
As long as $p-1\geq n,$ $n$ users can be supported.
Let $S=\{x_1,\ldots,x_n\}$ be the points that determine the currently used shares.
The "leftover shares" can be used later if new people join, so it may not be a bad idea to have $p$ appreciably bigger than $n,$ the current number of users.
Note that we assume that the dealer is a trusted third party and that the users will actually supply the correct share when asked by the dealer, otherwise things won't work, and more sophisticated schemes are needed.
This also applies if $k$ users can themselves get together to reconstruct, they must not lie about their shares and if exactly one of $k$ users is dishonest, he/she can learn the shares of the rest of the users, which means the rest cannot compute the secret correctly but he/she can if the remaining $k-1$ users are honest.