  The mergers of double neutron stars (DNSs) systems are believed to drive the
majority of short $\gamma$-ray bursts (SGRBs), while also serving as production
sites of heavy r-process elements. Despite being key to i) confirming the
nature of the extragalactic SGRBs, ii) addressing the poorly-understood
r-process enrichment in the ultra-faint dwarf galaxies (UFDGs), and iii)
probing the formation process of DNS systems, the space velocity distribution
of DNSs is still poorly constrained due to the small number of DNSs with
well-determined astrometry. In this work, we determine new proper motions and
parallaxes of two Galactic DNSs -- PSR J0509+3801 and PSR J1930-1852, using the
Very Long Baseline Array, and estimate the transverse velocities $v_\perp$ of
all the 11 isolated Galactic DNSs having proper motion measurements in a
consistent manner. Our correlation analysis reveals that the DNS $v_\perp$ is
tentatively correlated with three parameters: spin period, orbital
eccentricity, and companion mass. With the preliminary $v_\perp$ distribution,
we obtain the following findings. Firstly, the refined $v_\perp$ distribution
is confirmed to agree with the observed displacements of the localized SGRBs
from their host galaxy birth sites. Secondly, we estimate that around 11% and
25% of DNSs remain gravitationally bound to UFDGs with escape velocities of
15$\mathrm{~km~s^{-1}}$ and 25$\mathrm{~km~s^{-1}}$, respectively. Hence, the
retained DNSs might indeed be responsible for the r-process enrichment
confirmed so far in a few UFDGs. Finally, we discuss how a future ensemble of
astrometrically determined DNSs may probe the multimodality of the $v_\perp$
distribution.
