I have attached my current implementation for RANDOM_INIT.

For programs compiled without -fcoarry= or with -fcoarray=single,
the one gets, 

% cat random_init_2.f90
program foo
   real x(2)
   call random_init(.false., .false.)
   call random_number(x)
   print *, x
   call random_init(.false., .false.)
   call random_number(x)
   print *, x

   call random_init(.true., .false.)
   call random_number(x)
   print *, x
   call random_init(.true., .false.)
   call random_number(x)
   print *, x
end program foo
% gfcx -o z random_init_2.f90 && ./z
  0.817726076      0.318128884    
  0.598739505       2.99510360E-02
  0.336736381      0.870776474    
  0.336736381      0.870776474  

Now, with -fcoarray=lib, one gets 

%  gfcx -fcoarray=lib -c random_init_2.f90
f951: Fatal Error: RANDOM_INIT with co-arrays is broken!
compilation terminated.

I have zero knowledge about co-arrays and especially zero
knowledge about gfortran internals for co-arrays.  I'm
disinclined to waste another 12 hours trying to get gfortran
to emit essentially a call to this_image().  See iresolve.c
for details.

2018-01-07  Steven G. Kargl  <kargl@gcc.gnu.org>

	* check.c (gfc_check_random_init): New function.
	* gfortran.h: Define GFC_ISYM_RANDOM_INIT.
	* intrinsic.c (add_subroutines): Add random_init to list of subroutines.
	(gfc_check_intrinsic_standard): Update error message for Fortran 2018.
	* intrinsic.h: Add prototypes for gfc_check_random_init and
	gfc_resolve_random_init.
	* iresolve.c (gfc_resolve_random_init): Implementation.

2018-01-07  Steven G. Kargl  <kargl@gcc.gnu.org>

	* libgfortran/gfortran.map: Add _gfortran_random_init.
	* libgfortran/intrinsics/random.c: Add implemention of
	_gfortran_random_init

-- 
Steve