This is one of those little things that has always confused me. If someone said to you "in quantum mechanics, the eigenfunctions of a free particle are $\exp(ipx/\hbar)$" how would you know that this is the position space wavefunction, and not the momentum space one? The wavefunction itself is an explicit function of both.
There are a great many other simple quantum systems (square well, harmonic oscillator and so on) in which the momentum doesn't enter the real space wavefunction at all, but those particles still have an associated momentum.
So I guess my question is: what does that $p$ actually mean?