Here I would like to expand some of the arguments given in Ron Maimon's nice answer.
i) Let us divide the 1D $x$-axis into three regions $I$, $II$, and $III$, with a localized potential $V(x)$ in the middle region $II$ having a compact support. (Clearly, there are physically relevant potentials that hasn't compact support, e.g. the Coulomb potential, but this assumption simplifies the following discussion.)
ii) Time-independent and monochromatic. The particle is free in the regions $I$ and $III$, so we can solve the time-independent Schrödinger equation
$$\hat{H}\psi(x) ~=~E \psi(x), \qquad\qquad
\hat{H}~=~ \frac{\hat{p}^2}{2m}+V(x),\qquad\qquad E> 0, \qquad\qquad (1)$$
exactly there. We know that the 2nd order linear ODE has two linearly independent solutions, which in the free regions $I$ and $III$ are plane waves
$$ \psi_{I}(x) ~=~ a^{+}_{I}(k)e^{ikx} + a^{-}_{I}(k)e^{-ikx},
\qquad\qquad k> 0,
\qquad\qquad (2) $$
$$ \psi_{III}(x) ~=~ a^{+}_{III}(k)e^{ikx} + a^{-}_{III}(k)e^{-ikx},
\qquad\qquad (3) $$
Just from linearity of the Schrödinger equation, even without solving the middle region $II$, we know that the four coefficients $a^{\pm}_{I/III}(k)$ are constrained by two linear conditions. This observation leads, by the way, to the time-independent notion of the scattering $S$-matrix and the transfer $M$-matrix
$$ \begin{pmatrix} a^{-}_{I}(k) \\ a^{+}_{III}(k) \end{pmatrix}~=~ S(k) \begin{pmatrix} a^{+}_{I}(k) \\ a^{-}_{III}(k) \end{pmatrix}.
\qquad\qquad (4) $$
$$ \begin{pmatrix} a^{+}_{III}(k) \\ a^{-}_{III}(k) \end{pmatrix}~=~ M(k) \begin{pmatrix} a^{+}_{I}(k) \\ a^{-}_{I}(k) \end{pmatrix}.
\qquad\qquad (5) $$
see e.g. Griffiths' book, Introduction to Quantum Mechanics, Section 2.7, and this answer.
iii) Time-dependence of monochromatic wave. The dispersion relation reads
$$ \frac{E(k)}{\hbar} ~\equiv~\omega(k)~=~\frac{\hbar k^2}{2m},
\qquad\qquad (6) $$
The specific form $(6)$ of the dispersion relation will not matter in what follows. The full time-dependent monochromatic solution in the free regions I and III becomes
$$ \Psi_r(x,t)
~=~ \sum_{\sigma=\pm}a^{\sigma}_r(k)e^{\sigma ikx-i\omega(k)t}
~=~\underbrace{e^{-i\omega(k)t}}_{\text{phase factor}}
\Psi_r(x,0), \qquad r ~\in~ \{I, III\}. \qquad (7) $$
The solution $(7)$ is a sum of a right mover ($\sigma=+$) and a left mover ($\sigma=-$). For now the words right and left mover may be taken as semantic names without physical content. The solution $(7)$ is fully delocalized in the free regions I and III with the probability density $|\Psi_r(x,t)|^2$ independent of time $t$, so naively, it does not make sense to say that the waves are right or left moving, or even scatter. However, it turns out, we may view the monochromatic wave $(7)$ as a limit of a wave packet, and obtain a physical interpretation in that way, see next section.
iv) Wave packet. We now take a wave packet
$$ A^{\sigma}_r(k)~=~0 \qquad \text{for} \qquad
|k-k_0| ~\geq~ \frac{1}{L},
\qquad\sigma~\in~\{\pm\}, \qquad r ~\in~ \{I, III\},\qquad (8) $$
narrowly peaked about some particular value $k_0$ in $k$-space,
$$|k-k_0| ~\leq~ K, \qquad\qquad (9)$$
where $K$ is some wave number scale, so that we may Taylor expand the dispersion relation
$$\omega(k)~=~ \omega(k_0) + v_g(k_0)(k-k_0) + {\cal O}\left((k-k_0)^2\right), \qquad\qquad (10) $$
and drop higher-order terms ${\cal O}\left((k-k_0)^2\right)$. Here
$$v_g(k)~:=~\frac{d\omega(k)}{dk}\qquad\qquad (11)$$
is the group velocity. The wave packet (in the free regions I and III) is a sum of a right and a left mover,
$$ \Psi_r(x,t)~=~ \Psi^{+}_r(x,t)+\Psi^{-}_r(x,t),
\qquad\qquad r ~\in~ \{I, III\},\qquad\qquad (12) $$
where
$$ \Psi^{\sigma}_r(x,t)~:=~ \int dk~A^{\sigma}_r(k)e^{\sigma ikx-i\omega(k)t},
\qquad\qquad\sigma~\in~\{\pm\}, \qquad\qquad r ~\in~ \{I, III\}, $$
$$ ~\approx~ e^{i(k_0 v_g(k_0)-\omega(k_0))t}
\int dk~A^{\sigma}_r(k)e^{ ik(\sigma x- v_g(k_0)t)}$$
$$~=~\underbrace{e^{i(k_0 v_g(k_0)-\omega(k_0))t}}_{\text{phase factor}}
~\Psi^{\sigma}_r(x-\sigma v_g(k_0)t,0).\qquad\qquad (13)$$
The right and left movers $\Psi^{\sigma}$ will be very long spread-out wave trains of sizes $\geq \frac{1}{K}$, but we are still able to identity via eq. $(13)$ their time evolution as just
a collective motion with group velocity $\sigma v_g(k_0)$, and
an overall time-dependent phase factor of modulus $1$, which is the same for the right and the left mover.
In the limit $K \to 0$, with $K >0$, the approximation $(10)$ becomes better and better, and we recover the time-independent monochromatic wave,
$$ A^{\sigma}_r(k) ~\longrightarrow ~a^{\sigma}_r(k_0)~\delta(k-k_0)\qquad \text{for} \qquad K\to 0. \qquad\qquad (14)$$
It thus makes sense to assign a group velocity to each of the $\pm$ parts of the monochromatic wave $(7)$, because it can understood as an appropriate limit of the wave packet $(13)$.
