  We present a study of the environments of 17 Lyman-$\alpha$ (Ly$\alpha$)
emitting galaxies (LAEs) in the reionisation era ($5.8 < z < 8$) identified by
JWST/NIRSpec as part of the JWST Advanced Deep Extragalactic Survey (JADES).
Unless situated in sufficiently (re)ionised regions, Ly$\alpha$ emission from
these galaxies would be strongly absorbed by neutral gas in the intergalactic
medium (IGM). We conservatively estimate sizes of the ionised regions required
to reconcile the relatively low Ly$\alpha$ velocity offsets ($\Delta
v_\text{Ly$\alpha$}<300\,\mathrm{km\,s^{-1}}$) with moderately high Ly$\alpha$
escape fractions ($f_\mathrm{esc,\,Ly\alpha}>5\%$) observed in our sample of
LAEs, suggesting the presence of ionised hydrogen along the line of sight
towards at least eight out of 17 LAEs. We find minimum physical `bubble' sizes
of the order of $R_\text{ion}\sim0.1$-$1\,\mathrm{pMpc}$ are required in a
patchy reionisation scenario where ionised bubbles containing the LAEs are
embedded in a fully neutral IGM. Around half of the LAEs in our sample are
found to coincide with large-scale galaxy overdensities seen in FRESCO at $z
\sim 5.8$-$5.9$ and $z\sim7.3$, suggesting Ly$\alpha$ transmission is strongly
enhanced in such overdense regions, and underlining the importance of LAEs as
tracers of the first large-scale ionised bubbles. Considering only
spectroscopically confirmed galaxies, we find our sample of UV-faint LAEs
($M_\text{UV}\gtrsim-20\,\mathrm{mag}$) and their direct neighbours are
generally not able to produce the required ionised regions based on the
Ly$\alpha$ transmission properties, suggesting lower-luminosity sources likely
play an important role in carving out these bubbles. These observations
demonstrate the combined power of JWST multi-object and slitless spectroscopy
in acquiring a unique view of the early Universe during cosmic reionisation via
the most distant LAEs.
