If the client sees the same report and/or receipt they will drop the shopper and inform other MSCs. I know, because I have commiserated with shoppers who have had this happen. Get written permission from both schedulers because not to do so may otherwise well violate several clauses in their ICAs. MSC managers ans schedulers have also reported at conferences, concerning repercussions to doubling up reports without express permission to even do both shops with different reports.
I wasn't looking to do anything unethical which is why I asked. Good point on not sharing one shop result with both companies and it sounds as if I can do two of the same shops at the same place as long as I treat them separately for each company.
Either way, the best practice would be to get consent, in writing, from the schedulers of BOTH MSCs. You do not, have to, and should not tell, each MSC who the other MSC is. (ICA rules, again.)
None of these shops have the same reports. I can have 2 MSC's sending me to the same shop, talk to the same person, but ask different questions. I have no problem with that.