I've seen this a few times lately and I wanted to make sure I get it. Most recently I saw it on a video with an animal's thoughts narrated. It said この人間がいると安心できない。I take this as meaning "When this human is here I can't feel relieved/I can't feel relieved if this human is around". So, ～がいると means something to the effect of "when x exists/if x exists"? Would this be the most common way to talk about "you" and specific people being around as well? Such as 君がいると and さくらがいると?
Your question contains more than a couple of grammar peculiarities. The meaning is "I feel insecure whenever this person is around.",
"この人間” (this person) is someone close to the speaker at that time, so that your phrase is mostly supposed to be uttered by someone in a group, while someone in the group is speaking or doing something.
It is a phrase that could be uttered by a girl who feels somewhat assaulted by somebody.
If I understood the question correctly:
Yes, (I don't know if most common, but) at least I think it does sound pretty natural if someone says e.g.: 君が[e.g.近くに]いると安心できない
As to the この人, at least I'd say that you esp. in the contexts as above, can also use it to refer to yourself, like in;