Depends. If I'm leveling, it'll die if:
1. Someone isn't camping for it.
2. If it's not a pet made to be a pet (Challenge tames)
3. If I'm pissed off at it. (Vyragosa. I always. ALWAYS gun for Vyragosa... Or, I used to :3).
4. I need the achieve or experience (I have so many alts... and rares give sooo much xp.)
5. The person in front of me failed / I got first tag (champions... I loled cause my mage, running, tagged it before a paladin who flew on top of it. I got tag, he got aggro, that just amused me because he was spit spamming me for it... I just let him kill it a bit for me in return. -.- Be a jerk, I don't have to do anything.)
6. I'm with my bf. He has a hunter, but he derps a bit on the rarity of the new rares. He wouldn't kill a Loque... like. Ever. Or any spirit beast. But the new rares aren't so safe from him. xD He has a rare spawn trigger finger and they're too easy to kill at level

So even if I told him to stop, it would be too late.
It won't die, but I might tag/take it and run off with it if:
1. I know someone (CRZ or MG) who wants it and is online.
2. I want it on my hunter and find it on an alt.
3. It's a spirit beast in general or a pet I don't want to see dead. I'll find it a home, somehow.
I never advertise in general because there are so many hunters there who honestly... don't deserve it. At all. And I don't feel bad in saying that. It's the truth. I wouldn't lose sleep if they never got the pet they were camping because they're such assholes. I err on the side of caution when I'm "saving" a pet because I assume that everyone, unless proven otherwise, is that asshole. :/ I don't want to help the bad guy. I kind of wait to see if people are going to be nice to me without me offering something before I give it to them unless I've given them a rare before/exchanged "contact" information (I'll friend you, you friend me, I'll tell you if I see this rare, you tell me if you see that rare or trade spawn timers, ect).
But yeah. I have a set of rules to determine how I react to seeing certain rares.