Answer
The angle between the 2 initial velocities is $120^{\circ}$.
Work Step by Step
Let the final direction of the velocity of both masses together after the collision be in the y-direction. Then, the x-values for the momentum of both masses cancel out.
The total momentum in the y direction would be $\sum_y \vec{p}=mv\cos(\theta)+mv\cos(\theta)=2mv\cos(\theta)$, where $\theta$ is the angle of the velocity above the vertical or y-axis.
Using conservation of momentum, the final momentum would be $(m+m)\frac{v}{2}=mv$. Since final momentum equals initial momentum, $2mv\cos(\theta)=mv$, so the value of $\theta$ would be $60^{\circ}$. Since $\theta$ is the angle from the y-axis, the angle between the 2 velocites would be $60^{\circ}+60^{\circ}=120^{\circ}$.
