The realities of mass production lead to an unavoidable fact - all of the finished copies of a lens coming off a production line are not equal. There are good copies and bad copies, all assembled from individual parts each of which meets its specifications. The more a manufacturer tries to reduce the price or to include more features for a given price, the more pressure there is to accept components at looser acceptance criteria, which leads to wider variation in finished products.
So I see one person post that his new Sigma 50-500 gives him fabulous sharpness and contrast, while another posts his new 50-500 gives him nothing but junk. Both are probably right - that's the reality of good and bad copies.
I said all of this to lead to a decision of mine. Because of the problem of "bad" copies of less expensive third party lenses, I decided I would only buy a longer lens (in the range you are considering) after I had put the lens on my camera body, taken a few shots with it, and found those shots to be acceptable. And I would only buy the one on my camera - I would not take another in a box.
So I took my 80-400 I've had for 8 years and went to a local store. They had no 50-500, but they had a Sigma 150-500. I set up my tripod outside the camera store's front door and took a couple of 80-400 shots using my ML-L3 remote. Then I put on the Sigma 150-500, focused on the same subject and took a couple of shots in the same way. The Sigma images were obviously inferior to my 80-400, no doubt about it, and I didn't buy the Sigma. I've seen photos taken with a Sigma 150-500 and posted to various forums showing excellent sharpness, far better than I saw from the copy I tried. That's why I wanted to try the lens before buying.
An alternative: Tamron makes a 200-500, but it has no VR or OS built in. I shoot everything on a tripod, so that wasn't important to me, and my direct comparison test found it to give me images equal to or better than my 80-400. I bought it.
However, now I find I use it less than the 80-400. I'm very much used to the 80-400 zoom range, and I use all of it. I also keep a 16-85 on a second camera body, so I have a range of 16-400 covered with no gaps. When I have the 200-500 mounted, I find myself with a gap between 85 and 200 that has proven to be a problem at times. You didn't say what other lenses you have and only talked about the longer end of the ranges of the Nikon and Sigma lenses, but the zoom range is something to consider in choosing. Based on reputation, I think you'll get the best image quality from a Nikon 300/4 with Nikon 1.4x TC, but that only offers two focal lengths, 300 and 420, no zoom, and nothing shorter than 300. That would leave an undesirable gap for me. YMMV.
I also agree with Dieter - the 80-400 has been around for a long time and there are lots of good used ones available. I would not pay $1600+ for a new one when I can get one graded as Excellent at KEH for $1000. And, if the 200-500 range meets your needs and VR isn't necessary, the Tamron can be had for under $1000 new.
I haven't heard of reliability issues with Sigma, but i haven't owned one. However, based on the production variation issue, I strongly recommend that you accept the idea of paying a bit more for the lens in a brick-and-mortar store compared to purchasing online, go to local stores and try lenses on your camera, and only buy after you've verified yourself that the copy you are buying gives you the image quality you expect.