OK, I am struggling a little with this. It strikes me that this kid isn't really accepted anywhere. At home, her relationship with her parents is so fraught that her dad jokes she is bipolar (pretty strong language, even in jest). At the neighbor's house, the mom sees her as a threat to her own kids and a potential future bully and even possibly a sexual predator (??really??). So why wouldn't she act up? Kids will generally live up to whatever expectations you have of them.
Immediate reading for you and the other parents: 1-2-3 Magic. What you have described of the neighbors' parenting style is completely counterproductive for this kind of kid -- you don't attempt to engage or explain things in the heat of the moment, you explain the expectations before and simply remove the kid from the situation when she crosses the line. And "hard to set and enforce boundaries"? Yeah, it's hella hard. It's also his damn job, so he needs to man up and figure out a way to do it. Inconsistent boundaries will create exactly this kind of behavior.
My biggest recommendation for you is to try to develop a little empathy for this girl. I firmly believe there are no inherently bad kids*; usually what produces bad behavior is a disconnect between the kind of parenting/environment the kid needs and what the parents can provide.** Which brings me back to the "bipolar" crack as Exhibit A. I guarantee you she knows her dad thinks she is wrong in some fundamental respect. And I guarantee you she knows you don't like her. Kids like this are usually hyper-sensitive to others' emotions -- she knows it, it hurts/scares her, and you can see the results. Thus, empathy: if she is acting like this on the outside, imagine how scary it is to be inside her own head! Kids fundamentally want love and acceptance, so how desperate must a kid be if she is intentionally triggering negative attention and punishment?
From my experience with a hyper, bossy, extremely verbal attention-hound: these kids are exhausting, like a black hole of need that can never be filled. The fundamental problem with parenting a kid like that is that they are "on" all the time, and so you learn to tune them out and ignore the little annoyances (either because you just need five. freaking. minutes without hearing the word "mommy," or because the kid is hair-trigger on the meltdowns and you can't handle that right at that moment). But your frustration continues to build ("why can't she just [insert "normal" kid behavior here]"), and so when she finally does something that you just can't take, you end up snapping and overreacting.
So think about that from the kid's perspective: she's doing her normal stuff, she's trying to get mommy's attention, but mommy is ignoring her, and maybe mommy seems a little annoyed, but she can't figure out why; and then a few minutes later she's doing the same stuff, and now mommy turns on her and is really angry/frustrated and she gets a big punishment. I can't even tell you how counterproductive that is. The kid cannot possibly predict the consequences of her actions; that makes her feel very, very insecure; and the natural response to insecurity is to try to assert even more control and become even bossier and more demanding. And of course she also learns that if she wants mommy's attention, she needs to misbehave.
The only things that worked for me were "Your Spirited Child" and "1-2-3 Magic." The former helped me look for and understand my DD's triggers and needs. And the latter helped me understand how my parenting was contributing to the problems -- I wasn't doing my kid any favors ignoring the little stuff; in fact, letting my frustration build until I reacted emotionally to her misbehavior was creating more insecurity in her that just made things worse. The book forced me to explain the rules and consequences when I was calm, give the warnings before I got angry/frustrated/annoyed with her, and then give the consequence without the anger. It created an environment of calm structure for DD, where she could predict the consequences of her actions with 100% certainty, which in turn allowed her to feel safe and accepted and relax a little bit. And it stopped providing her attention for misbehaving; I replaced that with lots and lots of attention for doing something good (e.g., "look at you, sitting right down to do your homework!").***
I am recommending these books as much for you as for the parents. Definitely recommend them to the mom/dad next time they complain (e.g., "a friend said . . . ."). But that kind of calm, proactive approach can really help her at your house, too. Sure, you can't "fix" her -- but you certainly can provide some of the calm, predictable structure and a feeling of acceptance that can help her calm down a bit and not need to control everything (and everyone) so desperately.
I also don't want to sound harsh or unsympathetic to you or the other parents. I had one of those kids, so I know exactly how overwhelming it is -- and honestly, none of the "normal" parenting advice works. Ignore a tantrum? Worked great with my DS; epic, colossal fail with my DD. But when you have spent your whole life believing you don't "reward" a kid for bad behavior, it's a huge shift to realize that your tantrum-thrower is overwhelmed and needs help learning how to calm herself down and in fact needs to be cuddled, not ignored.**** And then that frustration is compounded by the constant judgment of other parents with "normal" kids, who imply that you are causing the problems, and if you'd just ignore the tantrums, they'd go away. You (OP) come across as wanting to be helpful and empathetic to these parents, not as judgy or anything, so I am recommending "Your Spirited Child" particularly for you to give you some insight into how differently these kinds of kids perceive the world and what does and doesn't work, so maybe you can really be that empathetic ear for these parents even though you don't have one of those kids yourself.
*OK, maybe it's possible to be born a sociopath/psychopath. But if so, that is a very, very small percentage of kids. And it's not this one.
**Note that I do NOT mean any of these parents are "bad," or that managing this kind of kid is easy under any circumstances -- just that there are different parenting styles, and some kids connect with one style better than others, and some parents are better at adjusting their styles to their kids' needs than others.
***I admit, I struggled with this; it smacked of "everyone gets a trophy"; do I really want to praise a kid for doing what she knows she ought to do anyway (especially when she is quiet and I finally have five minutes in which she ISN'T demanding my attention)? Well, when your kid craves attention like oxygen, and you want her to do what she should do instead of what she shouldn't, the answer is an unmitigated "yes."
**** And even as I type this, part of my brain still says "Really?? Cuddle a tantrum??? WTF????" Even though doing just that stopped 2.5 years of tantrums within a couple of weeks. It is SO hard to ignore what you expect "good parenting" to be.