Distribution of randomness in elliptic curve Diffie-Hellman shared secret
Fix a short Weierstrass curve $y^2 = x^3 + a x + b$ over a finite field $\mathbb F_p$ of order $n$.  (secp224r1 is an example, with $p = 2^{224} - 2^{96} + 1$, $a = -3$, and $b$ selected from an allegedly randomly chosen but unexplained seed passed through SHA-1.)

There are at most $n$ distinct values of $(x, y) = [\alpha \beta]G$ where $\alpha$ and $\beta$ are secret scalars in $\mathbb Z/n\mathbb Z$ and $G$ is the standard base point, all satisfying this equation.

If you choose uniformly among them, e.g. by choosing $\alpha$ and $\beta$ independently uniformly at random, and encode the pair $(x, y)$ into a $2 \lceil\log_2 p\rceil$-bit string, there is a trivial algorithm to distinguish the resulting string from a uniform random $2 \lceil\log_2 p\rceil$-bit string: decode it as a pair of integers $x$ and $y$ and test whether $y^2 \equiv x^3 + a x + b \pmod p$.  So not only do the encoded values of $(x, y)$ cover a tiny fraction of the space of $2 \lceil\log_2 p\rceil$-bit strings—square root of the number of all such strings of that length—but they're trivial to distinguish from uniform random.

But that's partly because for each $x$, there's only at most two values of $\pm y$, so we can compress it into a much shorter $(1 + \lceil\log_2 p\rceil)$-bit string by storing only a bit to discriminate between the two possible values of $y$.  Is that uniform random?  No!  If you decode $x$, you can test whether $x^3 + a x + b$ is a quadratic residue modulo $p$, which only about half of the integers are.

Distinguishers for the encoding of uniform random elliptic curve points were some of the first problems documented publicly in Dual_EC_DRBG[1], before everyone realized it was an obvious vector for a back door.

In general, when a random string is readily distinguishable from uniform like this, it is unfit to be used as a secret key in a cryptosystem whose contract demands a uniform random key—if you violate the security contract, it is null and void.  But if the random string, while not uniform, nevertheless has high min-entropy, you can effectively get a uniform random string by hashing it, say with SHAKE256—what this means is that the best an adversary can do is test a guess about what the input might have been.

As for the low-order or high-order bits of $x$ and $y$?  Don't bother; just hash them all.  The cost of computing a hash like SHAKE256 is negligible compared to the cost of computing an elliptic curve scalar multiplication even on a small curve like secp224r1.