I finally found a way to obtain these velocities with calculus. I am sure that one can always try to deduce the particles velocities by mere logic; in fact, more than once, I "thought" about the answer that I finally found. But I was not comfortable without a calculus development.
Finding the velocity of particle 1
We can first calculate the velocity of the system center of mass by taking particle 1 as a reference.
\begin{equation} \vec{v}_c = \vec{v}_1 + \dot\theta\vec{k} \times \frac{l}{2}\vec{e}_r \end{equation}
where $\vec{v}_c$ is the system center of mass velocity; $\vec{v}_1$ is the total velocity of particle 1; $\dot\theta\vec{k}$ is the angular velocity around an axis perpendicular to the $(x,y)$ plane; $\times$ is the cross product operator ;and $\vec{e}_r$ is the radial vector along the massless rod.
We know that particle 1 cannot have a vertical velocity because of the assumption of an inelastic collision. 512
So the equation above becomes:
\begin{equation} \vec{v}c = \vec{v}{1x} + \dot\theta\vec{k} \times \frac{l}{2}\vec{e}_r \end{equation}
where $\vec{v}_{1x}$ is the horizontal velocity of particle 1. By developing the above equation, we end up with:
\begin{equation} \vec{v}c = \vec{v}{1x} + \frac{l\dot\theta\sqrt{2}}{4} \left( -\vec{i} + \vec{j} \right) \end{equation}
Now, if the system center of mass does not move horizontally, that means that the horizontal component of its velocity has to be zero. Consequently, in the above equation:
\begin{equation} \vec{v}_{1x} - \frac{l\dot\theta\sqrt{2}}{4}\vec{i} = 0 \end{equation}
and since the vertical velocity of particle 1 is simply zero, then:
\begin{equation} \vec{v}_{1} = \frac{l\dot\theta\sqrt{2}}{4}\vec{i} \end{equation}
Finding the velocity of particle 2
The velocity of particle 2 can now be simply calculated as:
\begin{equation} \vec{v}_2 = \vec{v}_1 + \dot\theta\vec{k} \times l\vec{e}_r \end{equation}
and by developing the above equation :
\begin{equation} \vec{v}_2 = \frac{l\dot\theta\sqrt{2}}{2} \left( -\frac{1}{2}\vec{i} + \vec{j} \right) \end{equation} 804