Shamir's Secret Sharing Scheme - prime security? Does the prime (which needs to be larger than the message) have to be chosen randomly for security, or can it just be a huge constant?

Also, the Wikipedia article for SSSS states


  The size of each share does not exceed the size of the original data.


but isn't each share comprised of both $X$ and $Y$? or is $X$ already known because of the prime? and if the prime is chosen at random then surely the prime has to be distributed with each share?
The prime can be just any prime, so it doesn't have to be chosen randomly (but of course it can). And the prime is made public, so it is not part of the shares. Public values are known by definition by everyone, therefore are not part of the shares.

But you're right, the quote is wrong about the size of the share, because it neglects $x$. However, it's really close, because $y$ is usually the same length as $p$ (uniform distributed), but $x$ can be chosen to be small.