No, please make it out to me or place small denomination, unmarked bills in a black satchel and leave in locker number 357 at the Greyhound Bus terminal in downtown LA. I will take the money to Mexico where it will be safe until you call for it. I'll send you my phone number as soon as I am safely out of the country with your money. Trust me.
California deposit checks should always be made out to the escrow company.
Every broker has their own procedures, but I never take a deposit check from a buyer, I ask them to hold keep it until the seller accepts their offer. At that point I ask the buyer to deliver the check directly to the escrow company themselves (unless they are an out-of-town buyer). As a buyer myself, I always follow this procedure regardless of what the broker I'm working with asks for. I look at it this way - my name, address, and account number are on that check. For security reasons, I want to make sure it touches as few hands and is seen by as few eyes as possible.