I'll write $T_f$ for the set of primes $p$ such that $f(x)$ is a bijection $\mathbb{F}_p \to \mathbb{F}_p$. I claim that $T_f$ is always either finite or else $\# \{ p \in T_f : p \leq y \} \sim c \tfrac{y}{\log y}$ for some $c>0$.

The following result is known as Schur's conjecture; a flawed proof was given by [Fried][1], with corrected versions by [Turnwald][2] and [Müller][3]:

<b>Theorem</b> Let $f(x) \in \mathbb{Z}[x]$. If $T_f$ is infinite, then $f(x)$ is a composition of linear polynomials and [Dickson polynomials][4]. 

One should note that the Dickson polynomial $D_n(x,0)$ is just $x^n$, so this includes the possibility of including monomials in our composition.

A composition of functions $\mathbb{F}_p \to \mathbb{F}_p$ will be bijective if and only if all the functions composed are bijective. Linear functions are always bijective; the monomial $D_n(x,0)=x^n$ is bijective iff $GCD(n,p-1)=1$ and, for $a \neq 0$, the Dickson polynomial $D_n(x,a)$ is bijective iff $GCD(n,p^2-1)=1$ or, equivalently, $GCD(n,p-1) = GCD(n,p+1)=1$. (This last statement is copied from Lemma 1.4 in Turnwald; I didn't check it.) In short, imposing that our composition is bijective imposes finitely many conditions on the residue class of $p$ modulo various integers.

If these modular conditions can be satisfied by infinitely many primes, then they are satisfied by $\sim c \tfrac{y}{\log y}$ primes $\leq y$, by the PNT in arithmetic progressions.


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=257033
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=1329867
  [3]: https://mathscinet.ams.org/mathscinet-getitem?mr=1429041
  [4]: https://en.wikipedia.org/wiki/Dickson_polynomial