  It is generally assumed that sources sending randomly two particles to one or
two different observers, named here random destination sources (RDS), cannot by
used for genuine quantum nonlocality tests because of the postselection
loophole. We demonstrate that Bell experiments not affected by the
postselection loophole may be performed with: (i) RDS and local postselection
using perfect detectors, (ii) RDS, local postselection, and fair sampling
assumption with any detection efficiency, and (iii) RDS and a threshold
detection efficiency required to avoid the detection loophole. These results
allow the adoption of RDS setups, which are more simple and efficient, for
long-distance free-space Bell tests, and extends the range of physical systems
which can be used for loophole-free Bell tests.
