  We investigate the X-ray properties of a large sample of 71 broad line and
narrow line AGN at 2<z<11 discovered by JWST in the GOODS fields, which have
the deepest Chandra observations ever obtained. Despite the widespread presence
of AGN signatures in their rest-optical and -UV spectra, the vast majority of
them is X-ray undetected. The stacked X-ray data of the non-detected sources
also results in a non-detection. The upper limit on the X-ray emission for many
of these AGN is one or even two orders of magnitude lower than expected from a
standard AGN SED. Heavy X-ray absorption by clouds with large (Compton thick)
column density and low dust content, such as the Broad Line Region (BLR)
clouds, can explain the X-ray weakness. In this scenario the BLR covering
factor should be much larger than in low-z AGN or luminous quasar; this is
supported by the larger equivalent width of the broad component of Halpha in
JWST-selected AGN. We also find that the JWST-discovered AGN lack the
prominent, fast outflows characterizing low-z AGN and luminous quasars,
suggesting that, in JWST-selected AGN, dense gas lingers in the nuclear region,
resulting in large covering factors. We also note that a large fraction of
JWST-selected AGN match the definition of NLSy1, typically characterized by a
steep X-ray spectrum, and this can further contribute to their observed
weakness at high-z. Finally, we discuss that the broad Balmer lines used to
identify type 1 AGN cannot be ascribed to Very Massive Stars, Tidal Disruption
Events, or Supernovae, although we show that a minority of the faintest broad
lines could potentially be associated with the echo of superluminous SNe or
TDE. Scenarios in which the broad lines are ascribed to galactic outflows are
also untenable. We emphasize that confirming any of the scenarios discussed
above will require X-ray missions more sensitive than Chandra. (abridged)
