We present spectroscopic follow-up of candidate luminous Ly{\alpha} emitters (LAEs) at $z=5.7-6.6$ in the SA22 field with VLT/X-SHOOTER. We confirm two new luminous LAEs at $z=5.676$ (SR6) and $z=6.532$ (VR7), and also present HST follow-up of SR6. These sources have luminosities L$_{\rm Ly\alpha} \approx 3\times10^{43}$ erg s$^{-1}$, very high rest-frame equivalent widths of EW$_0\gtrsim 200$ {\AA} and narrow Ly{\alpha} lines (200-340 km s$^{-1}$). VR7 is the most UV-luminous LAE at $z>6.5$, with M$_{1500} = -22.5$, even brighter in the UV than CR7. Besides Ly{\alpha}, we do not detect any other rest-frame UV lines in the spectra of SR6 and VR7, and argue that rest-frame UV lines are easier to observe in bright galaxies with low Ly{\alpha} equivalent widths. We confirm that Ly{\alpha} line-widths increase with Ly{\alpha} luminosity at $z=5.7$, while the Ly$\alpha$ lines of faint LAEs become broader at $z=6.6$, potentially due to reionization. We find a large spread of up to 3 dex in UV luminosity for $>L^{\star}$ LAEs, but find that the Ly{\alpha} luminosity of the brightest LAEs is strongly related to UV luminosity at $z=6.6$. Under basic assumptions, we find that several LAEs at $z\approx6-7$ have Ly{\alpha} escape fractions $\gtrsim100$ %, indicating bursty star-formation histories, alternative Ly$\alpha$ production mechanisms, or dust attenuating Ly$\alpha$ emission differently than UV emission. Finally, we present a method to compute $\xi_{ion}$, the production efficiency of ionising photons, and find that LAEs at $z\approx6-7$ have high values of log$_{10}(\xi_{ion}$/Hz erg$^{-1}) \approx 25.51\pm0.09$ that may alleviate the need for high Lyman-Continuum escape fractions required for reionization.