Newton's Third Law says that the force exerted on Magnet #1 by Magnet #2 is equal in magnitude to the force exerted on Magnet #2 by Magnet #1. Since they have the same mass, they will therefore accelerate at the same rate, and they will therefore meet exactly in the middle.
The relative strength of the magnets is a red herring. One way to see this is to note that the magnetic force between two magnetic dipoles is proportional to the product of the dipole moments similar to how the force between two charges is proportional to the product of the charges. Thus, the stronger magnet creates a stronger magnetic field at the location of the weaker magnet than vice versa; but the stronger magnet also responds more strongly to a given field than the weaker magnet does. These two factors cancel each other out, and the force ends up being the same.
This does neglect the effects of electromagnetic radiation, which could conceivably carry a small amount of momentum off to infinity. I would expect, from similar calculations involving electric dipoles, that the radiation would be proportional to the square of the jerk $j$. A bit more dimensional analysis shows that the net momentum flux at infinity due to the radiation fields should be something like
$$
\frac{d P_\text{rad}}{d t} \sim \frac{ \mu_0 m^2}{c^6} j^2.
$$
The radiation reaction force must therefore also be proportional to this quantity. If we denote $v$ as the velocity scale of the dipoles during their motion, $R$ as the length scale of the motion, and $T$ as the time scale, we have
$$
j \sim \frac{v}{T^2} \sim \frac{v}{(R/v)^2} \sim \frac{v^3}{R^2}
$$
and so the radiation reaction force on the dipoles should be (to within a few orders of magnitude
$$
\frac{d P_\text{rad}}{d t} \sim \frac{\mu_0 m^2}{R^4} \frac{v^6}{c^6}.
$$
Since the actual dipole force between the magnets will be roughly proportional to $\mu_0 m^2/R^4$, we conclude that the effects of the radiation reaction force will smaller than the those of the radiation reaction force by a factor of $(v/c)^6$ (to within a few orders of magnitude.) So long as the magnets remain non-relativistic, this effect would of course be utterly negligible.
