Don't know what SOP is there, but twice in the past when out a shootin on BLM land here in NV (85% of the state) I've met with local boys in blue (tan/khaki actually). Their first action usually involves calling in your car/truck tags getting some info there.
Those two times, we saw them approach, put things down before they got there and moseyed out to greet them. BS'd for a bit about what we were shooting, clean up (policing our brass plus other people/slobs trash etc)... t'were no big deal.
I figure they're doing their job. I know they do call in each stop they make and have certain procedures required whenever they do exit their car. I'm just as polite and professional to them as I expect them to be to me and these two guys were both.
It's the non-LEO's that stop nearby that get my attention and you darn sure better not be surprised when out ashootin in the desert. Just not a wise move. When people drive by, as they approach, shooting stops and they get a head nod or a wave. I might be smilin, but I am armed. (I assume they are too and they usually do smile or show their empty hands with a wave). Judging from the amount of trash a lot of BLM shooters seem to leave about, I typically prejudge them as being one of those trashy types. In 25+ years of BLM shooting, I've yet to meet one who picks up and takes out more target trash than they brought... a sad statement unto itself.
I cannot imagine sneaking up on someone who has a loaded gun in their hand either (from the LEO's POV, maybe she wasn't sneaking at all, just walked up to you amazed that you let her get so close). How close was the Deputy when you "first heard" them behind you? (besides too close)
Any Lessons Worth Learning?
I preach taking out more stuff than you brought even if it's only one piece of trash put there by someone else... two or more is better (the more lead we leave, the more trash we take. A fair trade off maybe).
NEVER let anyone come up on your 6 without you being aware of their presence. Never. (unless they're reeeeeeal goooood and sneaky... like Lone Watie (Chief Dan George) in the Outlaw Josey Wales
)
Never shoot all of your ammo. Leave at least one magazine or cylinder full for the drive out.
But that's just me. and I typically don't worry about or even comment on LEO encounters... unless they surprise me. And that'd be my fault.