Can an alternating magnetic field be "informationally" shielded?
can information be sent out from an electrically shielded environment
  via alternating magnetic fields?


Yes. 


  How effective is an electrical shield against information blocking
  that may be carried by magnetic waves?


Depending on the thickness of the shielding and the frequency, anywhere from very effective to totally useless. The thinner the shield and the lower the frequency, the less effective it is at attenuating the magnetic field. Whether it will be enough to 'block' the signal also depends on distance and sensitivity of the receiver, and the nature of the signal. 

Real shields don't completely block the electric field either. If you put a sensitive receiver close enough to a 'totally' shielded high frequency transceiver you could probably pick up enough rf to get some information from it.