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 both sources. These sources have luminosities L_Ly {alpha}_~=3x10^43^erg/s, very high rest-frame equivalent widths of EW_0_>=200{AA} and narrow Ly{alpha} lines (200-340km/s). 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 there are indications that 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* 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~=6-7 have Ly{alpha} escape fractions >=100 per cent, 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 ionizing photons, and find that LAEs at z~=6-7 have high values of log_10_({xi}_ion_/Hz/erg)~=25.51+/-0.09 that may alleviate the need for high Lyman-Continuum escape fractions required for reionization.