If you are worried about a hit elk running onto someone else's property, you could approach the neighbors beforehand. Let them know who you are, where you will be hunting, educate them that even a perfectly hit animal that dies within 30 seconds may run quite a ways. You could find out if they want to be informed that one ran onto their property before you go retrieve it or that they don't care since you've already contacted them. You might also let them know that if they do not want the gutpile left there that you would be willing to remove it. Of course, you can also let them know that a lot of birds and animals benefit from scavenging the gutpile.
This proactive approach may gain you more access as well, unless the neighbors all hate the guy whose property you are on....
I would also not ask for permission to recover an animal. Be polite but act like it is expected to be able to recover the animal.
Good luck.