How much time do you have to spend on your editing? Conceptually, you could edit video on whatever old technology a piece of software that can edit 1080p will run. I know iMovie 06 will run on circa 2004 Macs... maybe earlier ones too, and it can edit 1080p.
BUT, the old hardware will slow down the pace of editing: wait for import, wait for each edit to render, wait for export. In other words, just about everyone is right based upon what you've heard. What's being left out is how fast you want to do the editing. If it's a one-off edit, maybe just be patient and make it work on slow tech. However, if you plan on regularly editing 1080p, then the waiting will weigh on you.
For example, I shoot a lot of 1080p. If I do the editing and render it on very new & fast hardware, I can render a couple of hours of it in a few hours- let's say 3 hours. If I transfer the master file over to- say- a 2004 Powerbook, it will render the file too but it might take a day or two before it gets done with that render. 3 hours or 40 hours, what's important to you?
What you leave out of your question is the value of your own time. If it has no value, then you've asked a perfect question (minimum specs will mean maximum time for editing & rendering). If you qualified your question by showing some value for your time, then you'll get suggestions for better hardware. If you make time very important, then you'll get best specs recommendations. What happens when you ask it as you have is that each person will inject their own value of their own time and give you an answer that fits their needs. That's why you get many different answers.
If you plan to regularly edit 1080p video and your time has some value, get the best hardware you can afford. If you don't mind burning a lot of time, the minimum specs will do the trick. I have a 2004 PowerMac G4 that can edit & render 1080p very, very SLOWLY. I doubt it's actually "minimum specs" but it's near that. If you want to buy it, PM me and I'll quote you a price.