  We investigate the ionizing properties of the pair of bright Ly$\alpha$
emitting galaxies BDF521 and BDF2195 at z=7.012 in order to constrain their
contribution to the formation of the BDF "reionized bubble". We obtain
constraints on UV emission lines (CIV$\lambda 1548$ doublet, HeII$\lambda
1640$, OIII]$\lambda 1660$ doublet, and CIII]$\lambda 1909$ doublet) from deep
VLT-XSHOOTER observations and compare them to those available for other
high-redshift objects, and to models with mixed stellar and AGN emission. We
use this spectroscopic information together with the photometry available in
the field to constrain the physical properties of the two objects using the
spectro-photometric fitting code BEAGLE. We do not detect any significant
emission at the expected position of UV lines, with 3$\sigma$ upper limits of
EW$\lesssim$2-7 AA rest-frame. We find that the two objects have lower CIII]
emission than expected on the basis of the correlation between the Ly$\alpha$
and CIII] EWs. The EW limits on CIV and HeII emission exclude pure AGN
templates at $\sim2-3\sigma$ significance, and only models with a $\lesssim$40%
AGN contribution are compatible with the observations. The two objects are
found to be relatively young ($\sim$20-30 Myrs) and metal-poor ($\lesssim 0.3
Z_{\odot}$) with stellar masses of a few $10^9M_{\odot}$. Their production rate
of hydrogen ionizing photons per intrinsic UV luminosity is
log($\xi_{ion}^*$/Hz erg$^{-1}$)=25.02-25.26, consistent with values typically
found in high-redshift galaxies, but more than twice lower than values measured
in $z>$7 galaxies with strong CIII] and/or optical line emission
($\simeq$25.6-25.7). The two BDF emitters have no evidence of higher than
average ionizing capabilities and are not capable of reionizing their
surroundings by their own means under realistic assumptions on the escape
fraction of ionizing photons. (abridged)
