There is an article on todays google news search on Mormons about a lady seeking help from police in Georgia because male priesthood Mormons have harassed her to a frightening level.
She is a campaigner for women to be ordained to the priesthood.

If she goes to a highly mormon populated area like Utah,then can she expect full protection from angry Mormon men aggrieved at her campaign for the priesthood?Many local police would be Mormon priesthood men.Would they have more sympathy for Mormon men aggrieved at her campaign for women to have the priesthood,or would they earnestly protect her from any angry Mormons making threats against her?

Police in Athens,Georgia have advised her to inform Utah police before going there to campaign.What can be expected in such a circumstance?

They assume the police won't let them file a report on Mormon harassment and will always take the side of morg stalkers.

Trespassing is trespassing. Come to the door with a bat.

There appears to be some confusion about police obligation here.

Police aren't obligated to protect *anybody.*

See Warren vs. DC for the horrifying background:
http://en.wikipedia.org/wiki/Warren_v._District_of_Columbia

Getting to the case at hand, the woman was harassed via email and Internet. Link is here (OP didn't post a link):
http://onlineathens.com/blotter/2015-05-21/woman-concerned-her-safety-after-speaking-out-about-mormon-faith

Internet and email trolling simply don't rise to the level of warranting immediate police action unless the writings contain specific and credible threats. Most of the time, they do not. There was no indication anybody had ever been to her house or spoke to her in person.

Answering the question directly, in order for Mormon police to ignore revenge against apostates, 3 things would need to occur:

-- The LDS church would need some kind of institutional messaging mechanism to let LEO's know who the targets are;
-- The LDS church would need some kind of institutional mechanism to instruct its members to target said individuals;
-- LEO's would have to actually allow said targeting to occur on their watch and do nothing.

The reality is, Utah's a pretty safe place, and Mormons are a generally (at least, statistically speaking) non-violent people with regards to murder and other forcible felonies. There's no church-sanctioned targeting of apostates going on. I'm sure Steve Benson and Tom Philips have received their fair share of Internet harassment, but I've not yet read or heard about any kind of targeting of them physically.

So, to answer your question, this woman's not entitled to any protection, anywhere, unless she buys it or brings it with her. Even in Utah, she will receive police response and/or protection in accordance with the level of threat. Online harassment usually doesn't merit protection for anybody.

There is no LDS institutional program in effect for Mormon LEO's to allow targeting of apostates. I was Mormon when I was in both Utah corrections and federal law enforcement, and I was certainly never brought in on anything of the type. If anything else, it'd make the church look bad if an apostate were harmed or killed, so the church would want to do everything in its power to prevent it.

And before somebody mentions the Danites, yes, I've heard of them. They went the way of the dodo 100 years past.

But what if the dodo goes the way of the Danite? Then we're all in trouble.

This question is absolutely insane. I don't think she has anything to worry about. Maybe in some other paranoid version of reality.

You better not have been one of the people I'm remembering posting about ferguson.

Pretty sure you have to get a restraining order and have to prove you are threatened. You can also tell them to leave. If they don't then you have reason to call the police. All they'll do is tell them to stay away.

The bat thing I posted was more of a joke. But they probably wouldn't come back if you did that.

The biggest danger to apostates from mormons is that their dysfunctional thinking could put you in an unsafe situation if there is conflict. Under those circumstances, they'll be likely to back eachother up on a story involving you that is not true, to make any mis-deeds on their part be covered up. For example, if you are gay, and there is conflict involving you , you might be assaulted and they'll all agree that you threw the first punch. I had an ex-girlfriend living next to me. After things didn't work out, she had several of my friends and neighbors over for sex. I could hear it on the other side of my bedroom wall. I told everyone's church leaders and there were several church courts held as a result. All of them hated me immensely. I was later set up for a malicious prosecution. Several of them said things in the police report that weren't true. A few Bishops even found out about the malicious prosecution and instead of having their person confess what they did to the prosecutor (as I insisted they do), they were only told by their church leaders to not testify, leaving me to either accept a plea bargain or to face a trial.

The point is that under conditions where everything is sweetness and light, you're probably safe around mormon church members. But if you expose their mis-deeds or anger them for whatever reason, you'll be dealing with people who have no ethics, and who might even believe that god is on their side as they commit injustices against you.